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A *power* is an algebraic or numeric expression that consists of a *base* and an *exponent*. In this lesson, exponential notation will be extended to rational numbers.
### Catch-Up and Review

Simplify.

**Here are a few recommended readings before getting started with this lesson.**

**Here are a few practice exercises before getting started with this lesson.**

Calculate the value of the following expressions. Write the answer as an integer number or as a fraction in its simplest form.

a $72 (3)$

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b $144 $

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c $(a )_{2}$

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d $3a_{3} $ {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["a"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["a"]}}

e Use the Power of a Power Property to simplify the expression $(a_{3})_{3}.$

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Challenge

The exponent of an expression indicates how many times the $base$ is multiplied by itself.

Power | Multiplication | Value |
---|---|---|

$4_{2}$ | $4⋅4$ | $16$ |

$2_{3}$ | $2⋅2⋅2$ | $8$ |

$5_{4}$ | $5⋅5⋅5⋅5$ | $625$ |

$1_{5}$ | $1⋅1⋅1⋅1⋅1$ | $1$ |

Discussion

By definition, to calculate the value of a power, the base is multiplied by itself as many times as indicated by the exponent. So, what happens with expressions where the exponent is a rational number, such as $2_{31}?$ It might be confusing to multiply $2$ by itself $31 $ times.
If the Power of a Power Property holds true for rational exponents, then $2_{31}$ must be a number that, when raised to the third power, equals $2.$ Such a number already exists: $32 .$

$2_{31}=2×? $

Considering the Power of a Power Property, notice what happens when the expression $2_{31}$ is raised to the third power.
$(2_{31})_{3}$

▼

Simplify

$2$

$(32 )_{3}=2 $

Therefore, for this property to still work, $2$ raised to the power of $31 $ is defined as the cube root of $2.$ $2_{31}=32 $

Discussion

The relationship between rational exponents and roots can be extended to *any* rational exponent of the form $n1 ,$ where $n$ is a natural number.

Concept

For any real number $a$ and natural number $n,$ the expression $a_{n1}$ is defined as the $n_{th}$ root of $a.$ Note that a root with an even index is defined only for non-negative numbers. Therefore, if $n$ is even, then $a$ must be non-negative.

Example

Dominika pays $625_{41}$ dollars per hour to study with a private tutor.

As her first homework assignment, she was given the task of writing how much she pays per hour as a radical. Write her answer without including the currency symbol.

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Pop Quiz

Calculate the values of the given powers with rational exponents by using a radical.

Example

The length of the diagonal is given to an accuracy of six decimal digits, the greatest known computational accuracy in the ancient world. Using the Pythagorean Theorem, calculate the

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Consider a right triangle in which the legs are two sides of the square and the hypotenuse is the diagonal.

Consider the right triangle formed by two sides of the square and its diagonal.

Here, the length of both legs is $1.$ Therefore, by substituting $a=1$ and $b=1$ into the Pythagorean Theorem, the length of the diagonal $c$ can be found.$a_{2}+b_{2}=c_{2}$

SubstituteII

$a=1$, $b=1$

$1_{2}+1_{2}=c_{2}$

▼

Solve for $c$

BaseOne

$1_{a}=1$

$1+1=c_{2}$

AddTerms

Add terms

$2=c_{2}$

SqrtEqn

$LHS =RHS $

$2 =c_{2} $

SqrtPowToNumber

$a_{2} =a$

$2 =c$

RearrangeEqn

Rearrange equation

$c=2 $

Discussion

Note that the numerator of a rational number does not have to be $1,$ but could also be *any* integer number. Therefore, rational numbers with a numerator different than $1$ should also be considered as rational exponents.

Concept

When a number is raised to the power of a fraction, that fraction is the number's rational exponent. Such an expression is equivalent to a root.

Notice that the denominator of the rational exponent gives the index of the root, while the numerator gives the power to which $a$ is raised. Since $n$ is a denominator, it cannot be zero. Moreover, if $n$ is an even number, then $a_{m}$ must be non-negative.

Discussion

Using the Power of a Power Property, another expression equivalent to $a_{nm}$ involving a radical can be found.
Therefore, $a_{nm}$ can also be defined as $(na )_{m}.$ Note that if $n$ is an even number, then $a$ must be non-negative. In conclusion, there are two definitions for $a_{nm}.$

$a_{nm}$

▼

Simplify

MoveNumRight

$ba =b1 ⋅a$

$a_{n1⋅m}$

ProdInExponent

$a_{m⋅n}=(a_{m})_{n}$

$(a_{n1})_{m}$

PowToRootD

$a_{n1}=na $

$(na )_{m}$

$a_{nm}=na_{m} ora_{nm}=(na )_{m} $

Example

Ignacio ate some cake at a birthday party. When he arrived home, he told his parents that he had $8_{32}$ slices.

His mother, who is a math teacher, asked him to express the number of slices that he ate as a radical.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\sqrt[3]{64}","\\sqrt[3]{8^2}","\\sqrt[3]{8}^2","(\\sqrt[3]{8})^2"]}}

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To calculate the value of $8_{32},$ it should be first rewritten as a radical. In order to do that, the definition of a rational exponent can be used.
Now Ignacio is able to give an answer for his mother. He can say that he ate $364 $ slices of the cake. To answer his father, Ignacio needs to find the value of the found radical.

Discussion

Now, focus on roots with an even index. For example, consider $3 .$ By definition, $3 $ is a number that, when raised to the second power, equals $3.$
*all* the numbers that are equal to $3$ when raised to the second power? There are two such numbers, $3 $ and $-3 .$
*positive* number that, when raised to the $nth$ power, equals $a.$

$(3 )_{2}=3 $

What about $x_{2}=3⇒x=3 orx=-3 $

To avoid complicating the definitions of $a_{n1}$ and $a_{nm},$ positive $3 $ is conventionally defined as the principal root. Therefore, for any even number $n,$ $na $ is defined as the Closure

It is important to know how to write expressions with rational exponents as radicals. Sometimes it is required to simplify an expression by using only radicals. Consider the following example.
### Hint

### Solution

Moreover, it is important to know how to write radical expressions using rational exponents, since rational and integer exponents have the same properties. Simplifying expressions involving exponents may be much easier than simplifying expressions involving radicals. Consider one last example.
### Hint

### Solution

$Simplify the expression andrewrite it using radicals only.3_{53}⋅2_{51} $

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Start by writing both numeric expressions using radicals.

Start by writing both factors in the expression using radicals.

$3_{53}⋅2_{51}$

$a_{nm}=na_{m} $

$53_{3} ⋅2_{51}$

PowToRootD

$a_{n1}=na $

$53_{3} ⋅52 $

MultRoot

$5a ⋅5b =5a⋅b $

$53_{3}⋅2 $

CalcPow

Calculate power

$527⋅2 $

Multiply

Multiply

$554 $

$Simplify the expression andrewrite it using exponents only.2_{31}⋅2 $

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Write $2 $ using a rational exponent and then use the Product of Powers Property.

The expression can be simplified by writing $2 $ using a rational exponent and then using the Product of Powers Property.

$2_{31}⋅2 $

SqrtToPowD

$a =a_{21}$

$2_{31}⋅2_{21}$

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$2_{31+21}$

ExpandFrac

$ba =b⋅2a⋅2 $

$2_{62+21}$

ExpandFrac

$ba =b⋅3a⋅3 $

$2_{62+63}$

AddFrac

Add fractions

$2_{65}$

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