A radical expression can be written in an equivalent form using a fractional (rational) exponent instead of a radical sign. It is often easier to write and to manipulate an expression with a fractional exponent than one written in radical form.

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Fractional (rational) exponents are an alternate way to express radicals. If x is a real number and m and n are positive integers:
exfracrule
 
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The denominator of the fractional exponent becomes the index (root) of the radical. The numerator of the fractional exponent becomes the power of the value under the radical symbol OR the power of the entire radical.
Why TWO
possible results?
If we apply the rules of exponents, we can see how there are two possible ways to convert an expression with a fractional exponent into an expression in radical form.
rule2

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The exponent of one half means the second root, or the
square root: square1.
The exponent of one third means the third root, or the
cube root: square3.
And so on ...


FracExRuleBox1

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Remember:
All RULES that apply to exponents, also apply to fractional exponents!


bullet Convert from Exponential to Radical Form:
convert2

Remember the denominator of the fractional exponent will become the root of the radical, and the numerator will become the power.

Create the power first and then the root.

OR

Create the root first and then the power.

Either way, you will have a correct answer.

 

bullet Convert from Radical to Exponential Form:
Remember the index (root) of the radical will become the denominator of the fractional exponent, and the power will become the numerator.
Create the denominator first and then the numerator.
OR
Create the numerator first and then the denominator.
Either way, you will have a correct answer.
fracconvertbb
Notice in the last example, that raising a square root to a power of 2 removes the radical.
Squaring and square rooting are inverse operations. One undoes the other.





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When simplifying radicals, it is often easier to find the answer by first rewriting the radical with fractional exponents. Let's see two examples:
1. exam2
2. exam3


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expin1
ex1
 
 

This problem will deal with a cube root and a square. The cube root of -64 is -4.
ex2



expin2
ex3
 
  Address the "negative" portion of the exponent first by inverting the fractional base. Then deal with the square root and the power of 3.
ex4



expin3 ex5
 

First, apply the exponent rule of raising a power to a power. Then deal with the roots and powers. At some point, replace a with 8.
ex6



expin4 ex7
  Replace x with 9, deal with the "negative" value of the exponent, and deal with the root.
ex8


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