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.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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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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b
Write sqrt(2^3) using a rational exponent.
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c
Write 3^(45) as a radical expression.
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Challenge
Rewriting an Algebraic Expression
Consider the following algebraic expression. sqrt(8x^3) * (x^(12)y)^5/sqrt(2x^2y)* x^(13)
Can the above expression be written using only rational exponents? Using only radicals?
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Discussion
Properties of Exponents
The following table shows the Properties of Exponents.
| Product of Powers Property | a^m a^n = a^(m+n) |
|---|---|
| Quotient of Powers Property | a^m/a^n= a^(m-n) |
| Power of a Product Property | (ab)^n=a^nb^n |
| Power of a Quotient Property | (a/b)^n = a^n/b^n |
| Power of a Power Property | (a^m)^n=a^(m* n) |
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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.
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=p/r and n=q/r
Using the above definitions, it can be proven that a^m* a^n is equal to a^(m+n).
Recall that p and q are integer numbers. Therefore, the Product of Powers Property can be used. sqrt(a^p * a^q) = sqrt(a^(p+q)) The resulting expression can be written using only exponents.
It has been proven that a^m* a^n=a^(m+n) for rational numbers m and n.
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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.
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=p/r and n=q/r
Using the above equalities, a^ma^n can be proven to be equal to a^(m-n).
Recall that p and q are integer numbers. Therefore, the Quotient of Powers Property can be used. sqrt(a^p/a^q) = sqrt(a^(p-q)) The resulting expression can be written using only exponents.
It has been proven that a^ma^n=a^(m-n) for rational numbers m and n.
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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.
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=p/q
Using the above equality, (ab)^n can be proven to be equal to a^nb^n.
Recall that p is an integer number. Therefore, the Power of a Product Property can be used to rewrite the expression. sqrt((ab)^p) = sqrt(a^pb^p) 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.
sqrt(a^pb^p)
RootProd
sqrt(a* b)=sqrt(a)*sqrt(b)
sqrt(a^p) sqrt(b^p)
sqrt(a^m)=a^(mn)
a^(pq) b^(pq)
Substitute
p/q= n
a^n b^n
It has been proven that (ab)^n=a^nb^n for rational n.
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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.
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=p/q
Using the above equality, it can be proven that ( ab)^n is equal to a^nb^n.
Recall that p is an integer number. Therefore, the Power of a Quotient Property can be used to rewrite the expression. sqrt((a/b)^p) = sqrt(a^p/b^p) 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.
sqrt(a^p/b^p)
RootQuot
sqrt(a/b)=sqrt(a)/sqrt(b)
sqrt(a^p)/sqrt(b^p)
sqrt(a^m)=a^(mn)
a^(pq)/b^(pq)
Substitute
p/q= n
a^n/b^n
It has been proven that ( ab)^n= a^nb^n for rational n.
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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.
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. m=p/r and n=q/r
Using the above definitions, it can be proven that (a^m)^n is equal to a^(m* n).
Recall that a^(mn) can be defined either as sqrt(a^m) or (sqrt(a))^m. Using the second definition, the last obtained expression can be rewritten. (sqrt(a^p))^(qr) = (sqrt(sqrt(a^p)))^q The root of a root can be expressed using only one root whose index is the product of the original indices. (sqrt(sqrt(a^p)))^q ⇔ (sqrt(a^p))^q Note that since r is an integer number, then r* r is also integer. Using the fact that sqrt(a^m) and (sqrt(a))^m represent the same expression a^(mn), the power q can be moved inside the radical. (sqrt(a^p))^q ⇔ sqrt((a^p)^q) As p and q are integer numbers, the Power of a Power Property can be used.
sqrt((a^p)^q)
PowPow
(a^m)^n=a^(m* n)
sqrt(a^(p* q))
sqrt(a^m)=a^(mn)
a^(p* qr* r)
WriteProdFrac
Write as a product of fractions
a^(pr* qr)
SubstituteII
p/r= m, q/r= n
a^(m* n)
It has been proven that (a^m)^n=a^(m* n) for rational numbers m and n.
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Example
Simplifying a Numeric Expression
Classmates Vincenzo and Magdalena have each simplified the same numeric expression 3^(14)* 3^(12). Yet, they obtained different results.
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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.
3^(34)
a^(mn)=sqrt(a^m)
sqrt(3^3)
Vincenzo's answer is also correct! Therefore, both Magdalena and Vincenzo are correct.
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Example
Simplifying the Powers in an Algebraic Expression
Dylan was asked to simplify the expression x^(12) y^(27) * x^(13) y^(37) 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.
x^(12) y^(27) * x^(13) y^(37)
CommutativePropMult
Commutative Property of Multiplication
x^(12) x^(13)* y^(27) y^(37)
MultPow
a^m*a^n=a^(m+n)
x^(12+ 13)* y^(27+ 37)
▼
Add fractions
ExpandFrac
a/b=a * 3/b * 3
x^(36+ 13) * y^(27+ 37)
ExpandFrac
a/b=a * 2/b * 2
x^(36+ 26) * y^(27+ 37)
AddFrac
Add fractions
x^(56) * y^(57)
Multiply
Multiply
x^(56) y^(57)
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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 2^(56)÷ 2^(15). Once again, they obtained different results!
{"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.
2^(56)÷ 2^(15)
DivToFrac
a÷ b=a/b
2^(56)/2^(15)
DivPow
a^m/a^n= a^(m-n)
2^(56- 15)
▼
Subtract fractions
ExpandFrac
a/b=a * 5/b * 5
2^(2530- 15)
ExpandFrac
a/b=a * 6/b * 6
2^(2530- 630)
SubFrac
Subtract fractions
2^(1930)
Since Vincenzo's answer is 2^(16), he is not correct. To see if Magdalena is correct, the obtained expression needs to be rewritten as a radical.
2^(1930)
a^(mn)=sqrt(a^m)
sqrt(2^(19))
Since Magdalena wrote sqrt(2^5), she is not correct. Therefore, neither Vincenzo nor Magdalena obtained the correct answer this time.
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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. 2^(23)x^(12)* 2^(311)/2^(1011)x^(29) Help Dylan to complete the given task and then write the answer using rational exponents.
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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.
2^(23)x^(12)* 2^(311)/2^(1011)x^(29)
CommutativePropMult
Commutative Property of Multiplication
2^(23)2^(311)* x^(12)/2^(1011)* x^(29)
MultPow
a^m*a^n=a^(m+n)
2^(23+ 311)* x^(12)/2^(1011)* x^(29)
▼
Add fractions
ExpandFrac
a/b=a * 11/b * 11
2^(2233+ 311)* x^(12)/2^(1011)* x^(29)
ExpandFrac
a/b=a * 3/b * 3
2^(2233+ 933)* x^(12)/2^(1011)* x^(29)
AddFrac
Add fractions
2^(3133)* x^(12)/2^(1011)* x^(29)
WriteProdFrac
Write as a product of fractions
2^(3133)/2^(1011)* x^(12)/x^(29)
DivPow
a^m/a^n= a^(m-n)
2^(3133- 1011) * x^(12- 29)
▼
Subtract fractions
ExpandFrac
a/b=a * 3/b * 3
2^(3133- 3033) * x^(12- 29)
ExpandFrac
a/b=a * 9/b * 9
2^(3133- 3033) * x^(918- 29)
ExpandFrac
a/b=a * 2/b * 2
2^(3133- 3033) * x^(918- 418)
SubFrac
Subtract fractions
2^(133)* x^(518)
Multiply
Multiply
2^(133) x^(518)
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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":"x^(12)y^(34)\/x^(13)"},{"id":1,"text":"sqrt(x)* x^(15)"},{"id":2,"text":"sqrt(2)x* 2y\/2^(32)"},{"id":3,"text":"2x^(13)sqrt(y)\/sqrt(2)y^(14)2^(12)"}],[{"id":0,"text":"x^(16)y^(34)"},{"id":1,"text":"x^(710)"},{"id":2,"text":"xy"},{"id":3,"text":"x^(13)"}]],"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. x^(12)y^(34)/x^(13) Here, the Quotient of Powers Property for Rational Exponents can be used.
x^(12)y^(34)/x^(13)
MovePartNumRight
a* b/c=a/c* b
x^(12)/x^(13) * y^(34)
DivPow
a^m/a^n= a^(m-n)
x^(12- 13) y^(34)
▼
Subtract fractions
ExpandFrac
a/b=a * 3/b * 3
x^(36- 13) y^(34)
ExpandFrac
a/b=a * 2/b * 2
x^(36- 26) y^(34)
SubFrac
Subtract fractions
x^(16) y^(34)
Simplifying the Second Expression
Consider now the second expression. sqrt(x) * x^(15) 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.
sqrt(x) * x^(15)
SqrtToPowD
sqrt(a)=a^(12)
x^(12) x^(15)
MultPow
a^m*a^n=a^(m+n)
x^(12+ 15)
▼
Add fractions
x^(710)
Simplifying the Third Expression
Now, consider the third expression. sqrt(2)x* 2y/2^(32) 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.
sqrt(2)x* 2y/2^(32)
SqrtToPowD
sqrt(a)=a^(12)
2^(12)x* 2y/2^(32)
CommutativePropMult
Commutative Property of Multiplication
2^(12)2* xy/2^(32)
a=a^1
2^(12)2^1* xy/2^(32)
MultPow
a^m*a^n=a^(m+n)
2^(12+1)* xy/2^(32)
2^(32)* xy/2^(32)
MovePartNumRight
a* b/c=a/c* b
2^(32)/2^(32)* xy
▼
Simplify
xy
Simplifying the Fourth Expression
Finally, consider the last expression. 2x^(13)sqrt(y)/sqrt(2)y^(14)2^(12) Similarly to the previous expression, this one can be simplified by using the definition of a rational exponent and different properties of rational exponents.
2x^(13)sqrt(y)/sqrt(2)y^(14)2^(12)
▼
Write radicals as powers
2x^(13)y^(14)/2^(12)y^(14)2^(12)
CommutativePropMult
Commutative Property of Multiplication
2x^(13)* y^(14)/2^(12)2^(12)* y^(14)
MultPow
a^m*a^n=a^(m+n)
2x^(13)* y^(14)/2^(12+ 12)* y^(14)
AddFrac
Add fractions
2x^(13)* y^(14)/2^1 * y^(14)
WriteProdFrac
Write as a product of fractions
2x^(13)/2^1* y^(14)/y^(14)
MovePartNumRight
a* b/c=a/c* b
2/2^1* x^(13)* y^(14)/y^(14)
▼
Simplify
ExponentOne
a^1=a
2/2* x^(13)* y^(14)/y^(14)
QuotOne
a/a=1
1* x^(13) * 1
IdPropMult
Identity Property of Multiplication
x^(13)
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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.
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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=SA^(12)/2π^(12)
b
V=SA^(32)/6π^(12)
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=4π r^2
sqrt(SA/4π)=r
SqrtToPowD
sqrt(a)=a^(12)
(SA/4π)^(12)=r
PowQuot
(a/b)^m=a^m/b^m
SA^(12)/(4π)^(12)=r
PowProdII
(a b)^m=a^m b^m
SA^(12)/4^(12)π^(12)=r
a^(12)=sqrt(a)
SA^(12)/sqrt(4)π^(12)=r
CalcRoot
Calculate root
SA^(12)/2π^(12)=r
RearrangeEqn
Rearrange equation
r=SA^(12)/2π^(12)
Note that when solving the equation, the principal root was taken. This is because the radius r must be a positive value.
b
In the formula for volume, substitute the expression for r obtained in Part A and simplify as much as possible.
V=4/3π r^3
Substitute
r= SA^(12)/2π^(12)
V=4/3π ( SA^(12)/2π^(12))^3
PowQuot
(a/b)^m=a^m/b^m
V=4/3π (SA^(12))^3/(2π^(12))^3
PowProdII
(a b)^m=a^m b^m
V=4/3π * (SA^(12))^3/2^3(π^(12))^3
CalcPow
Calculate power
V=4/3π * (SA^(12))^3/8(π^(12))^3
PowPow
(a^m)^n=a^(m* n)
V=4/3π (SA^(12* 3)/8π^(12* 3))
MoveRightFacToNumOne
1/b* a = a/b
V=4π/3 (SA^(32)/8π^(32))
▼
Multiply
MultFrac
Multiply fractions
V=4π SA^(32)/24π^(32)
ReduceFrac
a/b=.a /4./.b /4.
V=π SA^(32)/6π^(32)
CommutativePropMult
Commutative Property of Multiplication
V=SA^(32)π/6π^(32)
WriteProdFrac
Write as a product of fractions
V=SA^(32)/6(π/π^(32))
a=a^1
V=SA^(32)/6(π^1/π^(32))
DivPow
a^m/a^n= a^(m-n)
V=SA^(32)/6(π^(1- 32))
V=SA^(32)/6(π^(- 12))
NegExponentToFrac
a^(- m)=1/a^m
V=SA^(32)/6(1/π^(12))
MultFrac
Multiply fractions
V=SA^(32)/6π^(12)
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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. sqrt(8x^3) * (x^(12)y)^5/sqrt(2x^2y)* x^(13) Simplify the expression and write the answer using only rational exponents.
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Solution
Start by rewriting the radicals as powers with rational exponents.
sqrt(8x^3) * (x^(12)y)^5/sqrt(2x^2y)* x^(13)
SqrtToPowD
sqrt(a)=a^(12)
(8x^3)^(12) (x^(12)y)^5/sqrt(2x^2y)* x^(13)
RootToPowD
sqrt(a)=a^(1n)
(8x^3)^(12) (x^(12)y)^5/(2x^2y)^(13) x^(13)
Now, the properties studied in this lesson can be used to simplify the expression.
(8x^3)^(12) (x^(12)y)^5/(2x^2y)^(13) x^(13)
PowProdII
(a b)^m=a^m b^m
8^(12)(x^3)^(12) (x^(12))^5y^5/2^(13)(x^2)^(13)y^(13) x^(13)
Rewrite
Rewrite 8 as 2^3
(2^3)^(12)(x^3)^(12) (x^(12))^5y^5/2^(13)(x^2)^(13)y^(13) x^(13)
PowPow
(a^m)^n=a^(m* n)
(2^(3* 12)) (x^(3* 12)) (x^(12* 5))y^5/2^(13) (x^(2* 13))y^(13) x^(13)
▼
Multiply
MoveLeftFacToNumOne
a* 1/b= a/b
2^(32) x^(32) x^(12* 5)y^5/2^(13) x^(23)y^(13) x^(13)
MoveRightFacToNumOne
1/b* a = a/b
2^(32) x^(32) x^(52)y^5/2^(13) x^(23)y^(13) x^(13)
CommutativePropMult
Commutative Property of Multiplication
2^(32) x^(32) x^(52)y^5/2^(13) x^(23) x^(13) y^(13)
MultPow
a^m*a^n=a^(m+n)
2^(32) (x^(32+ 52))y^5/2^(13) (x^(23+ 13)) y^(13)
2^(32) x^4 y^5/2^(13) x^1 y^(13)
WriteProdFrac
Write as a product of fractions
2^(32)/2^(13) * x^4/x^1* y^5/y^(13)
DivPow
a^m/a^n= a^(m-n)
(2^(32- 13)) (x^(4-1)) (y^(5- 13))
▼
Subtract terms
SubTerms
Subtract terms
(2^(32- 13)) x^3 (y^(5- 13))
ExpandFrac
a/b=a * 3/b * 3
(2^(96- 13)) x^3 (y^(5- 13))
ExpandFrac
a/b=a * 2/b * 2
(2^(96- 26)) x^3 (y^(5- 13))
NumberToFrac
a = 3* a/3
(2^(96- 26)) x^3 (y^(153- 13))
SubFrac
Subtract fractions
2^(76) x^3 y^(143)
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Extending the Properties of Exponents to Rational Exponents
Exercise 3.1
Consider the following identities. b^A =MN, b^(C3) =M, b^(D4) =N Write an equation that shows the relationship between A, C, and D. Give the answer in the simplest form and without fractions.
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In order to write an equation that shows the relationship between A, C, and D, we have to eliminate the variables b, M, and N. Consider the identities we have that describe M and N in terms of b, C, and D. b^(C3)=M and b^(D4)=N This means that we can replace M and N with b^(C3) and b^(D4), respectively, in the equation b^A=MN. Then we can use the Product of Powers Property for Rational Exponents to simplify the equation even further. This property states that if we have a product of two powers with the same base, we add the rational exponents and leave the base unchanged.
The powers on both sides of the equation have the same base, which means that we can equate the exponents.
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How many 5^2-terms should be inside the parentheses on the left-hand side in order for the equation to be true? (5^2+...+5^2)^(12)=5^2+5^2+5^2
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Let's start by recalling that repeated addition can be written as multiplication. Let n be the number of terms of the addition in parenthesis. 5^2+...+5^2_n = n* 5^2 By using this identity, we can rewrite the given equation. (5^2+...+5^2)^(12)=5^2+5^2+5^2 ⇕ (n* 5^2)^(12)=5^2+5^2+5^2 We will now solve the equation for n. To do so, we can first use the Product of Powers Property for Rational Exponents. This property allows us to write the power of a product as a product of two powers.
Next, we can use the Power of a Power Property for Rational Exponents. According to this property, if we have the power of a power, we multiply the exponents and leave the base unchanged.
We can now finish solving the equation for n.
There must be 225 5^2-terms in the parentheses in order for the equation to be true.
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Extending the Properties of Exponents to Rational Exponents
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