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Rational exponents and radicals are two different ways to express the same quantities. Sometimes, one form is more useful than the other. Thus, being able to translate between the two is important.

$58_{2} $

RootToPowD

$na =a_{n1}$

$(8_{2})_{51}$

PowPow

$(a_{m})_{n}=a_{m⋅n}$

$8_{2⋅51}$

MoveLeftFacToNumOne

$a⋅b1 =ba $

$8_{52}$

$5x andx_{32}$

Show Solution *expand_more*

The properties of radicals allow expressions with radicals to be rewritten.

The nth root of a product can be written as the product of the nth root of each factor. For example, $37⋅9 ,$ can be expressed as follows using this rule.
This rule can be explained by expressing the radical using a rational exponent, and then using the Power of a Product Property.
This rule is valid for all values of a, b and n. However, in order to avoid non-real solutions it is necessary that a≥0 and b≥0 for even values of n.

$37⋅9 $

RootToPowSL

$na =a_{1/n}$

$(7⋅9)_{1/3}$

PowProd

$(a⋅b)_{m}=a_{m}⋅b_{m}$

$7_{1/3}⋅9_{1/3}$

PowToRootSL

$a_{1/n}=na $

$37 ⋅39 $

The nth root of a fraction can be written as the nth root of the numerator divided by the nth root of the denominator. For example, $397 ,$ can be expressed as follows using this rule.
This rule can be explained by rewriting the radical as a rational exponent, and then using the Power of a Quotient Property.
This rule is valid for all values of a and n, and when b≠0. However, in order to avoid non-real solutions it is necessary that a≥0 and b>0 for even values of n.

$397 $

RootToPowSL

$na =a_{1/n}$

$(97 )_{1/3}$

PowQuot

$(ba )_{m}=b_{m}a_{m} $

$9_{1/3}7_{1/3} $

PowToRootSL

$a_{1/n}=na $

$39 37 $

Show Solution *expand_more*

When two radicals with the same index are either multiplied or divided they can be simplified using the properties of radicals. We'll begin by simplifying the numerator using the Product Property of Radicals.
Next we simplify the expression using the Quotient Property of Radicals.
Thus, the expression simplifies to 3.

To simplify an $n_{th}$ root, it is necessary that the radicand can be expressed as a power. If the index of the radical and the power of the radicand are equal, $na_{n} ,$ the radical expression can be simplified as follows.

$na_{n} ={∣a,∣a∣, ifnis oddifnis even $

The absolute value of a number is always non-negative, so when n is even, the result will always be non-negative.

To write the expression for $na_{n} ,$ there are two cases to consider.

- a≥0
- a<0

Both cases will be considered one at a time.

In case of negative a, there are also two cases two consider.

- n is even
- n is odd

$na_{n} $

Simplify

NegBaseToPosPow

$(-a)_{n}=a_{n}$

$n(-a)_{n} $

RootToPowD

$na =a_{n1}$

$((-a)_{n})_{n1}$

PowPow

$(a_{m})_{n}=a_{m⋅n}$

$(-a)_{n⋅n1}$

$a⋅a1 =1$

(-a)1

ExponentOne

a1=a

-a

$ntimesy⋅y⋅…⋅y =a_{n} $

Because n is odd and a is negative, $a_{n}$ is also negative. This means that the best candidate for y is simply a. If a is non-negative, $na_{n} $ is always equal to a. However, in case of negative a, the value of $na_{n} $ depends on the *parity* of n.

a≥0 | a<0 | |
---|---|---|

Even n | $na_{n} =a$ | $na_{n} =-a$ |

Odd n | $na_{n} =a$ | $na_{n} =a$ |

To conclude, for odd n, the expression $na_{n} $ is equal to a. On the other hand, if n is even, $na_{n} $ can be written as ∣a∣.

$na_{n} ={∣a,∣a∣, ifnis oddifnis even $

Expressions with radicals can be written using rational exponents. Then, they can be simplified using the properties of exponents. Consider the following expression.
*expand_more*
*expand_more*
*expand_more*
*expand_more*
*expand_more*

1

Rewrite terms into $na_{n} $

When a term inside a radical has a power greater than the index of the radical, it can be rewritten into $na_{n} $-form. In the example, there are two such terms, $3x_{4} $ and $6x_{7} .$
The products under the radicals can now be rewritten using the Product Property of Radicals.

$4>3and7>6$

First, the Product of Powers Property can be used to rewrite the terms under these radicals.
$6x_{7} 4x_{3} ⋅3x_{4} $

Rewrite

Rewrite 4 as 3+1

$6x_{7} 4x_{3} ⋅3x_{3+1} $

Rewrite

Rewrite 7 as 6+1

$6x_{6+1} 4x_{3} ⋅3x_{3+1} $

SumInExponent

$a_{m+n}=a_{m}⋅a_{n}$

$6x_{6}⋅x_{1} 4x_{3} ⋅3x_{3}⋅x_{1} $

$6x_{6}⋅x_{1} 4x_{3} ⋅3x_{3}⋅x_{1} $

RootProd

$3a⋅b =3a ⋅3b $

$6x_{6}⋅x_{1} 4x_{3} ⋅3x_{3} ⋅3x_{1} $

RootProd

$6a⋅b =6a ⋅6b $

$6x_{6} ⋅6x_{1} 4x_{3} ⋅3x_{3} ⋅3x_{1} $

ExponentOne

a1=a

$6x_{6} ⋅6x 4x_{3} ⋅3x_{3} ⋅3x $

2

Simplify $na_{n} $-terms

The $na_{n} $-terms can now be simplified as follows.

$na_{n} ={∣a∣ifnis odd∣a∣ifnis even $

In the expression there are two terms that can be simplified using this rule.
$6x_{6} ⋅6x 4x_{3} ⋅3x_{3} ⋅3x $

RootPowToNumber

$3a_{3} =a$

$6x_{6} ⋅6x 4x_{3} ⋅x⋅3x $

$6a_{6} =∣a∣$

$∣x∣⋅6x 4x_{3} ⋅x⋅3x $

3

Express the radicals using rational exponents

Next, rewrite the radicals using rational exponents. The example can be rewritten as follows.

4

Simplify using the laws of exponents

When the expression is written using rational exponents, it can be simplified. When two terms with the same base are multiplied, the exponents are added according to the Product of Powers Property.
To simplify the exponent further, which requires adding and subtracting fractions, the denominators must be made the same. Here, the least common denominator is 12.
Since the expressions in the numerator and the denominator have the same bases, x, they can be simplified. First, by using the Quotient of Powers Property, the expression is written as one term. To simplify the exponent, the denominators must then be made the same.

$∣x∣⋅x_{61}x_{43}⋅x⋅x_{31} $

a=a1

$∣x∣⋅x_{61}x_{43}⋅x_{1}⋅x_{31} $

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$∣x∣⋅x_{61}x_{43+1+31} $

$∣x∣⋅x_{61}x_{43+1+31} $

ExpandFrac

$ba =b⋅3a⋅3 $

$∣x∣⋅x_{61}x_{129+1+31} $

OneToFrac

Rewrite 1 as $1212 $

$∣x∣⋅x_{61}x_{129+1212+31} $

ExpandFrac

$ba =b⋅4a⋅4 $

$∣x∣⋅x_{61}x_{129+1212+124} $

AddFrac

Add fractions

$∣x∣⋅x_{61}x_{1225} $

$∣x∣⋅x_{61}x_{1225} $

DivPow

$a_{n}a_{m} =a_{m−n}$

$∣x∣x_{1225−61} $

ExpandFrac

$ba =b⋅2a⋅2 $

$∣x∣x_{1213−122} $

SubFrac

Subtract fractions

$∣x∣x_{1211} $

5

Rewrite expression using radicals

When the expression has been simplified completely, it can be rewritten using radicals.
This is the simplified expression.

Show Solution *expand_more*

Notice that the terms in this expression each have a difference base.
Next, we can rewrite the expression using the Power of a Power Property.
All terms in the expression now have the same base. From here, we can use the Product of Powers Property and the Quotient of Powers Property to simplify.
The answer is $2_{-32},$ which can also be written as

$2,4,and16.$

This expression cannot be simplified until the bases are the same. Therefore, we can begin by rewriting the terms with base 2. Notice that 4=22 and 16=24.
$4_{34}⋅16_{31}2_{310} $

Rewrite

Rewrite 4 as 22

$(2_{2})_{34}⋅16_{31}2_{310} $

Rewrite

Rewrite 16 as 24

$(2_{2})_{34}⋅(2_{4})_{31}2_{310} $

$(2_{2})_{34}⋅(2_{4})_{31}2_{310} $

PowPow

$(a_{m})_{n}=a_{m⋅n}$

$2_{2⋅34}⋅2_{4⋅31}2_{310} $

Multiply

Multiply

$2_{38}⋅2_{34}2_{310} $

$2_{38}⋅2_{34}2_{310} $

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$2_{38+34}2_{310} $

AddFrac

Add fractions

$2_{312}2_{310} $

DivPow

$a_{n}a_{m} =a_{m−n}$

$2_{310−312}$

SubFrac

Subtract fractions

$2_{-32}$

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