2. Extending the Properties of Exponents to Rational Exponents
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Chapter 6
2.
Extending the Properties of Exponents to Rational Exponents
Rational exponents provide a bridge between roots and powers in mathematics. By understanding the properties of rational exponents, learners can handle complex expressions with ease. In practical scenarios, these properties offer shortcuts for simplifying calculations and streamlining problem-solving. Engineers, scientists, and researchers often leverage these properties to optimize algorithms, making computational tasks more efficient. Grasping these properties is thus a stepping stone to deeper insights and enhanced mathematical proficiency.
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| | 16 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Extending the Properties of Exponents to Rational Exponents
Slide of 16
In this lesson, it will be proven that the properties of exponents are also valid for rational exponents. Moreover, these properties will be used to simplify numeric and algebraic expressions with rational exponents and roots.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
- Product of Powers Property
- Quotient of Powers Property
- Power of a Power Property
- Power of a Product Property
- Power of a Quotient Property
- Rational Exponents
Here are a few practice exercises before getting started with this lesson.
a Use the properties of exponents for integer numbers to match each algebraic expression on the left with its equivalent expression on the right.
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class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.054008em;\"><span style=\"top:-3.3029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.897438em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">8<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u00f7<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":3,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">a<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u00f7<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathdefault\">b<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":4,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">a<\/span><span class=\"mord mathdefault\">b<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"}],[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">7<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":3,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.897438em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u00f7<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":4,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"}]],"lockLeft":false,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3,4],[0,1,2,3,4]]}
b Write 423 using a rational exponent.
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c Write 354 as a radical expression.
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Challenge
Rewriting an Algebraic Expression
Consider the following algebraic expression.
32x2y⋅x318x3⋅(x21y)5
Can the above expression be written using only rational exponents? Using only radicals?Discussion
Properties of Exponents
The following table shows the Properties of Exponents.
| Product of Powers Property | aman=am+n |
|---|---|
| Quotient of Powers Property | anam=am−n |
| Power of a Product Property | (ab)n=anbn |
| Power of a Quotient Property | (ba)n=bnan |
| Power of a Power Property | (am)n=am⋅n |
Discussion
Product of Powers Property for Rational Exponents
The multiplication of two powers with the same base a and rational exponents m and n results in a power with base a and exponent m+n.
The expression can be undefined for some non-positive values of a. Therefore, this rule will only be defined for positive values of a.
Proof
Product of Powers Property for Rational ExponentsSince m and n are rational numbers, they can be written as the quotients of two integers. Let p, q, and r be three integers such that r=0, m is the quotient of p and r, and n is the quotient of q and r.
m=rpandn=rq
Using the above definitions, it can be proven that am⋅an is equal to am+n.
Recall that p and q are integer numbers. Therefore, the Product of Powers Property can be used.
rap⋅aq=rap+q
The resulting expression can be written using only exponents.
It has been proven that am⋅an=am+n for rational numbers m and n. Discussion
Quotient of Powers Property for Rational Exponents
The quotient of two powers with same base a and rational exponents m and n results in a power with base a and exponent m−n.
The expression can be undefined for some non-positive values of a. Therefore, this rule will only be defined for positive values of a.
Proof
Quotient of Powers Property for Rational ExponentsSince m and n are rational numbers, they can be written as the quotients of two integers. Let p, q, and r be three integers such that r=0, m is the quotient of p and r, and n is the quotient of q and r.
m=rpandn=rq
Using the above equalities, anam can be proven to be equal to am−n.
Recall that p and q are integer numbers. Therefore, the Quotient of Powers Property can be used.
raqap=rap−q
The resulting expression can be written using only exponents.
It has been proven that anam=am−n for rational numbers m and n. Discussion
Power of a Product Property for Rational Exponents
Multiplying two numbers or variables and then raising the product to the power of n, where n is rational, is the same as raising the factors to the power of n and then multiplying them.
The expression can be undefined for some non-positive values of a and b. Therefore, this rule will only be defined for positive values of a and b.
Proof
Power of a Product Property for Rational ExponentsSince n is a rational number, it can be written as the quotient of two integers p and q, where q=0.
n=qp
Using the above equality, (ab)n can be proven to be equal to anbn.
Recall that p is an integer number. Therefore, the Power of a Product Property can be used to rewrite the expression.
q(ab)p=qapbp
The index of the above radical is an integer number, q. Therefore, the obtained expression can be rewritten as the product of two radicals. Then, the definition of a rational exponent can be used.
It has been proven that (ab)n=anbn for rational n. Discussion
Power of a Quotient Property for Rational Exponents
Dividing two numbers or variables and then raising the quotient to the power of n, where n is rational, is the same as raising the divisor and dividend to the power of n and then calculating the quotient.
The expression can be undefined for some non-positive values of a and b. Therefore, this rule will only be defined for positive values of a and b.
Proof
Power of a Quotient Property for Rational ExponentsSince n is a rational number, it can be written as the quotient of two integers p and q, where q=0.
n=qp
Using the above equality, it can be proven that (ba)n is equal to bnan.
Recall that p is an integer number. Therefore, the Power of a Quotient Property can be used to rewrite the expression.
q(ba)p=qbpap
The index q of the above radical is an integer number. Therefore, the obtained expression can be rewritten as the quotient of two radicals. Then, the definition of a rational exponent can be used.
It has been proven that (ba)n=bnan for rational n. Discussion
Power of a Power Property for Rational Exponents
If a power with base a and rational exponent m is raised to the power of n, where n is rational, then the result is a power with base a and exponent m⋅n.
The expression can be undefined for some non-positive values of a. Therefore, this rule will only be defined for positive values of a.
Proof
Power of a Power Property for Rational ExponentsSince m and n are rational numbers, they can be written as the quotients of two integers. Let p, q, and r be three integers such that r=0, m is the quotient of p and r, and n is the quotient of q and r.
It has been proven that (am)n=am⋅n for rational numbers m and n.
m=rpandn=rq
Using the above definitions, it can be proven that (am)n is equal to am⋅n.
Recall that anm can be defined either as nam or (na)m. Using the second definition, the last obtained expression can be rewritten.
(rap)rq=(rrap)q
The root of a root can be expressed using only one root whose index is the product of the original indices.
(rrap)q⇔(r⋅rap)q
Note that since r is an integer number, then r⋅r is also integer. Using the fact that nam and (na)m represent the same expression anm, the power q can be moved inside the radical.
(r⋅rap)q⇔r⋅r(ap)q
As p and q are integer numbers, the Power of a Power Property can be used.
r⋅r(ap)q
PowPow
(am)n=am⋅n
r⋅rap⋅q
nam=anm
ar⋅rp⋅q
WriteProdFrac
Write as a product of fractions
arp⋅rq
SubstituteII
rp=m, rq=n
am⋅n
Example
Simplifying a Numeric Expression
Classmates Vincenzo and Magdalena have each simplified the same numeric expression 341⋅321. Yet, they obtained different results. 
Use the Product of Powers Property for Rational Exponents to determine who is correct.
{"type":"choice","form":{"alts":["Vincenzo","Magdalena","Both","Neither"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}
Solution
The exponent of the product is the sum of the exponents of the factors. In the given expression, the exponents are fractions. To add two fractions and determine the exponent of the product, find equivalent fractions with the same denominator.
Magdalena's answer is correct. To see if Vincenzo's answer is correct, rewrite the obtained expression as a radical.
Vincenzo's answer is also correct! Therefore, both Magdalena and Vincenzo are correct.
343
anm=nam
433
Example
Simplifying the Powers in an Algebraic Expression
Dylan was asked to simplify the expression x21y72⋅x31y73 and write the answer using rational exponents for a homework assignment. However, Dylan did not pay attention during the lesson and now he has no clue how to find the answer. Use the Product of Powers Property for Rational Exponents and find the answer to help Dylan with his homework!
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Hint
Start by using the Commutative Property of Multiplication.
Solution
First, the Commutative Property of Multiplication can be used to rearrange the expression. Then, the Product of Powers Property can be applied.
x21y72⋅x31y73
CommutativePropMult
Commutative Property of Multiplication
x21x31⋅y72y73
MultPow
am⋅an=am+n
x21+31⋅y72+73
▼
Add fractions
ExpandFrac
ba=b⋅3a⋅3
x63+31⋅y72+73
ExpandFrac
ba=b⋅2a⋅2
x63+62⋅y72+73
AddFrac
Add fractions
x65⋅y75
Multiply
Multiply
x65y75
Example
Simplifying the Quotient of Two Powers
Vincenzo and Magdalena continue with their homework, and this time, they are asked to simplify the numeric expression 265÷251. Once again, they obtained different results!
Use the Quotient of Powers Property for Rational Exponents to determine who is correct.
{"type":"choice","form":{"alts":["Neither","Both","Vincenzo","Magdalena"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Solution
The exponent of the quotient is the difference between the exponents of the dividend and the divisor. In the given expression, the exponents are fractions. To subtract two fractions, find equivalent fractions with the same denominator.
Since Vincenzo's answer is 261, he is not correct. To see if Magdalena is correct, the obtained expression needs to be rewritten as a radical.
Since Magdalena wrote 625, she is not correct. Therefore, neither Vincenzo nor Magdalena obtained the correct answer this time.
23019
anm=nam
30219
Example
Simplifying a Rational Expression Containing Products and a Quotient
Dylan is making progress with his homework. In one of the exercises, he is asked to simplify the given expression.
21110x92232x21⋅2113
Help Dylan to complete the given task and then write the answer using rational exponents. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":true,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["2^{\\frac{1}{33}}x^{\\frac{5}{18}}","2^{\\frac{1}{33}}\\t x^{\\frac{5}{18}}","x^{\\frac{5}{18}}2^{\\frac{1}{33}}","x^{\\frac{5}{18}}\\t 2^{\\frac{1}{33}}"]}}
Hint
Start by using the Commutative Property of Multiplication. Then, use the Product of Powers Property for Rational Exponents. Finally, apply the Quotient of Powers Property for Rational Exponents.
Solution
First, the Commutative Property of Multiplication can be used to rearrange the expression. Then, the Product of Powers Property can be used to simplify the numerator. Finally, the Quotient of Powers Property can be used to fully simplify the expression.
21110x92232x21⋅2113
CommutativePropMult
Commutative Property of Multiplication
21110⋅x922322113⋅x21
MultPow
am⋅an=am+n
21110⋅x92232+113⋅x21
▼
Add fractions
ExpandFrac
ba=b⋅11a⋅11
21110⋅x9223322+113⋅x21
ExpandFrac
ba=b⋅3a⋅3
21110⋅x9223322+339⋅x21
AddFrac
Add fractions
21110⋅x9223331⋅x21
WriteProdFrac
Write as a product of fractions
2111023331⋅x92x21
DivPow
anam=am−n
23331−1110⋅x21−92
▼
Subtract fractions
ExpandFrac
ba=b⋅3a⋅3
23331−3330⋅x21−92
ExpandFrac
ba=b⋅9a⋅9
23331−3330⋅x189−92
ExpandFrac
ba=b⋅2a⋅2
23331−3330⋅x189−184
SubFrac
Subtract fractions
2331⋅x185
Multiply
Multiply
2331x185
Example
Matching Equivalent Rational Expressions
Dylan is trying to finish the last few exercises of his homework. Help him get a passing grade by matching the equivalent expressions!
{"type":"pair","form":{"alts":[[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.46059em;vertical-align:-0.82957em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.63102em;\"><span style=\"top:-2.17043em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.93957em;\"><span style=\"top:-3.3485500000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.82957em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.04em;vertical-align:-0.23972em;\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8002800000000001em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathdefault\">x<\/span><\/span><\/span><span style=\"top:-2.76028em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z'\/><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.23972em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.9540200000000001em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">5<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.41379em;vertical-align:-0.82957em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.5842200000000002em;\"><span style=\"top:-2.17043em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.93957em;\"><span style=\"top:-3.3485500000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.90722em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-2.86722em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z'\/><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.13278em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.82957em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"},{"id":3,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.7049700000000003em;vertical-align:-1.02401em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.68096em;\"><span style=\"top:-2.17043em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.90722em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-2.86722em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z'\/><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.13278em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.93957em;\"><span style=\"top:-3.3485500000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.93957em;\"><span style=\"top:-3.3485500000000004em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.7269399999999995em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord sqrt\"><span class=\"root\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.541892em;\"><span style=\"top:-2.719672em;\"><span class=\"pstrut\" style=\"height:2.5em;\"><\/span><span class=\"sizing reset-size6 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.70306em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><span style=\"top:-2.66306em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z'\/><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33693999999999996em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.02401em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"}],[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.14846em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">6<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9540200000000001em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">7<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><\/span><\/span><\/span>"},{"id":3,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9540200000000001em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9540200000000001em;\"><span style=\"top:-3.363em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mopen nulldelimiter sizing reset-size3 size6\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8443142857142858em;\"><span style=\"top:-2.656em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span style=\"top:-3.2255000000000003em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"><\/span><\/span><span style=\"top:-3.384em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.344em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter sizing reset-size3 size6\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}
Hint
Use and combine the properties of exponents presented in this lesson.
Solution
The properties of rational exponents can be combined to simplify algebraic or numeric expressions. Analyze each of the given expressions one at a time.
Simplifying the First Expression
Consider the first expression.x31x21y43
Here, the Quotient of Powers Property for Rational Exponents can be used.
x31x21y43
MovePartNumRight
ca⋅b=ca⋅b
x31x21⋅y43
DivPow
anam=am−n
x21−31y43
▼
Subtract fractions
x61y43
Simplifying the Second Expression
Consider now the second expression.x⋅x51
Here, the radical expression can be written as a power with a rational exponent. Then, the Product of Powers Property for Rational Exponents can be applied.
Simplifying the Third Expression
Now, consider the third expression.2232x⋅2y
Here, the definition of a rational exponent will be used again to rewrite the radical expression. Then, the Product of Powers Property for Rational Exponents and the Quotient of Powers Property for Rational Exponents will be used. 2232x⋅2y
SqrtToPowD
a=a21
223221x⋅2y
CommutativePropMult
Commutative Property of Multiplication
2232212⋅xy
a=a1
22322121⋅xy
MultPow
am⋅an=am+n
223221+1⋅xy
223223⋅xy
MovePartNumRight
ca⋅b=ca⋅b
223223⋅xy
▼
Simplify
xy
Simplifying the Fourth Expression
Finally, consider the last expression.2y412212x314y
Similarly to the previous expression, this one can be simplified by using the definition of a rational exponent and different properties of rational exponents.
2y412212x314y
▼
Write radicals as powers
221y412212x31y41
CommutativePropMult
Commutative Property of Multiplication
221221⋅y412x31⋅y41
MultPow
am⋅an=am+n
221+21⋅y412x31⋅y41
AddFrac
Add fractions
21⋅y412x31⋅y41
WriteProdFrac
Write as a product of fractions
212x31⋅y41y41
MovePartNumRight
ca⋅b=ca⋅b
212⋅x31⋅y41y41
▼
Simplify
ExponentOne
a1=a
22⋅x31⋅y41y41
QuotOne
aa=1
1⋅x31⋅1
IdPropMult
Identity Property of Multiplication
x31
Pop Quiz
Practicing the Power of a Power Property for Rational Expressions
Use the Power of a Power Property for Rational Exponents to calculate the exponent after simplifying the given expression. Please enter only the exponent into the answer box and write it as a fraction. Even if it is possible, do not simplify the fraction.
Note that the properties of rational exponents are helpful for almost every field in math.
Example
Using Properties of Rational Exponents to Rewrite Formulas
Below are the formulas for the surface area and the volume of a sphere with radius r.
a Express the radius r in terms of the surface area SA. Write the answer using rational exponents in the simplest form.
b Using the answer for Part A, express the volume V in terms of the surface area SA. Write the answer using rational exponents in the simplest form.
Answer
a r=2π21SA21
b V=6π21SA23
Hint
a Use inverse operations to isolate r in the surface area formula.
b In the volume formula, substitute the expression for r obtained in Part A. Then, simplify as much as possible.
Solution
a Use inverse operations to isolate r on one side of the equation.
SA=4πr2
4πSA=r
SqrtToPowD
a=a21
(4πSA)21=r
PowQuot
(ba)m=bmam
(4π)21SA21=r
PowProdII
(ab)m=ambm
421π21SA21=r
a21=a
4π21SA21=r
CalcRoot
Calculate root
2π21SA21=r
RearrangeEqn
Rearrange equation
r=2π21SA21
b In the formula for volume, substitute the expression for r obtained in Part A and simplify as much as possible.
V=34πr3
Substitute
r=2π21SA21
V=34π(2π21SA21)3
PowQuot
(ba)m=bmam
V=34π(2π21)3(SA21)3
PowProdII
(ab)m=ambm
V=34π⋅23(π21)3(SA21)3
CalcPow
Calculate power
V=34π⋅8(π21)3(SA21)3
PowPow
(am)n=am⋅n
V=34π(8π21⋅3SA21⋅3)
MoveRightFacToNumOne
b1⋅a=ba
V=34π(8π23SA23)
▼
Multiply
MultFrac
Multiply fractions
V=24π234πSA23
ReduceFrac
ba=b/4a/4
V=6π23πSA23
CommutativePropMult
Commutative Property of Multiplication
V=6π23SA23π
WriteProdFrac
Write as a product of fractions
V=6SA23(π23π)
a=a1
V=6SA23(π23π1)
DivPow
anam=am−n
V=6SA23(π1−23)
V=6SA23(π-21)
NegExponentToFrac
a-m=am1
V=6SA23(π211)
MultFrac
Multiply fractions
V=6π21SA23
Closure
Applying Properties of Exponents to Simplify an Expression
In this lesson, the understanding of the properties of exponents was extended to include rational exponents. Using these properties, the challenge presented at the beginning of the lesson can now be solved.
32x2y⋅x318x3⋅(x21y)5
Simplify the expression and write the answer using only rational exponents. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["2^{\\frac{7}{6}} x^3 y^{\\frac{14}{3}}","2^{\\frac{7}{6}} y^{\\frac{14}{3}} x^3","x^3 2^{\\frac{7}{6}} y^{\\frac{14}{3}}","x^3 y^{\\frac{14}{3}} 2^{\\frac{7}{6}}","y^{\\frac{14}{3}} x^3 2^{\\frac{7}{6}}","y^{\\frac{14}{3}} 2^{\\frac{7}{6}} x^3"]}}
Solution
Start by rewriting the radicals as powers with rational exponents.
Now, the properties studied in this lesson can be used to simplify the expression.
32x2y⋅x318x3⋅(x21y)5
SqrtToPowD
a=a21
32x2y⋅x31(8x3)21(x21y)5
RootToPowD
na=an1
(2x2y)31x31(8x3)21(x21y)5
(2x2y)31x31(8x3)21(x21y)5
PowProdII
(ab)m=ambm
231(x2)31y31x31821(x3)21(x21)5y5
Rewrite
Rewrite 8 as 23
231(x2)31y31x31(23)21(x3)21(x21)5y5
PowPow
(am)n=am⋅n
231(x2⋅31)y31x31(23⋅21)(x3⋅21)(x21⋅5)y5
▼
Multiply
231x32y31x31223x23x25y5
CommutativePropMult
Commutative Property of Multiplication
231x32x31y31223x23x25y5
MultPow
am⋅an=am+n
231(x32+31)y31223(x23+25)y5
231x1y31223x4y5
WriteProdFrac
Write as a product of fractions
231223⋅x1x4⋅y31y5
DivPow
anam=am−n
(223−31)(x4−1)(y5−31)
▼
Subtract terms
SubTerms
Subtract terms
(223−31)x3(y5−31)
ExpandFrac
ba=b⋅3a⋅3
(269−31)x3(y5−31)
ExpandFrac
ba=b⋅2a⋅2
(269−62)x3(y5−31)
NumberToFrac
a=33⋅a
(269−62)x3(y315−31)
SubFrac
Subtract fractions
267x3y314
Use the properties of rational exponents and properties of radicals to simplify the numeric expressions.
42⋅253
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["2^{\\frac{17}{20}}"]}}
473⋅372
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\sqrt[7]{576}"]}}
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We want to use the properties of rational exponents to simplify the given numeric expression. To do so, we will first rewrite the radical in the expression as a power with a rational exponent. Then, we can multiply the powers using the Product of Powers Property for Rational Exponents.
We found that the given expression simplifies to 2^(1720). sqrt(2)* 2^(35) = 2^(1720)
Once again we want to use the properties of rational exponents or properties of radicals to simplify the given numeric expression. To do so, we will start by rewriting both powers as radicals. Then we can use the Product Property of Radicals.
We found that the given expression simplifies to sqrt(576).
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Rewrite the numeric expression so that it contains only one radical.
532 521
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{1}{\\sqrt[6]{5}}"]}}
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Let's start by using the Quotient of Powers Property for rational exponents. This property allows us to write the numeric expression as a single power. According to this rule, if we have a quotient of two powers with the same base but different rational exponents, we subtract the exponents.
Next, we can use the Negative Exponent Property to avoid a negative exponent.
Finally, we can rewrite the power as a radical. To do so, recall that we can rewrite a power with a rational exponent whose numerator is 1 as a radical expression where the index is the denominator of the rational exponent and the radicand is the base of the power. Let's do it!
We found that the given numeric expression simplifies to 1sqrt(5).
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Simplify the expression as much as possible. Write the answer as a radical.
13-72⋅13721375
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\sqrt[7]{13}"]}}
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To simplify the fraction, we will start by using the Quotient of Powers Property for Rational Exponents. This property allows us to write the second factor of the numeric expression as a single power. According to this rule, if we have a quotient of two powers with the same base but different rational exponents, we subtract the exponents.
Next, we will use the Product of Powers Property for Rational Exponents. According to this rule, if we have a product of two powers with the same base but different rational exponents, we add the exponents.
Finally, we will rewrite the power with a rational exponent as a radical. To do so, recall that we can rewrite a power with a rational exponent whose numerator is 1 as a radical expression where the index is the denominator of the rational exponent and the radicand is the base of the power. Let's do it!
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Evaluate the numeric expression.
(72964)61
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{2}{3}"]}}
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To evaluate the given numeric expression, we will first use the Power of a Quotient Property for Rational Exponents. This property allows us to convert the powers of one quotient to the quotient of two powers.
Next, we will rewrite both the numerator and denominator as radicals. To do so, recall that we can rewrite a power with a rational exponent whose numerator is 1 as a radical expression where the index is the denominator of the rational exponent and the radicand is the base of the power.
Finally, we can evaluate the numerator and denominator by writing both radicands as powers of 6, then canceling out the exponents of the radicands with the index of the roots.
We found that the given expression simplifies to 23.
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Simplify the algebraic expressions.
(x6)31
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["x^2"]}}
(x43y32)61
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["x^{\\frac{1}{8}} y^{\\frac{1}{9}}","y^{\\frac{1}{9}}x^{\\frac{1}{8}}","\\sqrt[8]{x}\\sqrt[9]{y}"]}}
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To simplify the given algebraic expression, we will use the Power of a Power Property for Rational Exponents. According to this property, if we have a power of another power, we multiply the exponents and leave the base unchanged.
We found that the given expression simplifies to x^2.
To simplify this expression, we will start by using the Power of a Product Property for Rational Exponents. This property allows us to have the product of two powers instead of having the power of a product.
Next, we can use the Power of a Power Property for Rational Exponents. This property states that if we have the power of a power, we can rewrite the expression by multiplying the exponents and leaving the base unchanged. Let's do it!
We found that the given expression simplifies to x^(18)y^(19).
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Extending the Properties of Exponents to Rational Exponents
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