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Fractional Exponents and Radicals

Fractional Exponents and Radicals

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Author: Sophia Tutorial
Description:

Write fractional exponents as radicals, or vice versa.

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what's covered
  1. Inverse Operations: Radicals and Exponents
  2. Fractional Exponents and Radicals

1. Inverse Operations: Radicals and Exponents

Just as multiplication and division are inverse operations of one another, radicals and exponents are also inverse operations. For example, suppose we have the the number 3 and we raise it to the second power. Now if we were to take the square root of 32, notice that we will end up with 3. This is the number with which we started.

square root of 3 squared end root space equals space 3

Below are a few more examples:

cube root of 4 cubed end root space equals space 4

fourth root of 2 to the power of 4 end root space equals space 2

root index 9 of 8 to the power of 9 space end root space equals space 8

square root of 5 squared end root space equals space 5

2. Fractional Exponents and Radicals

Whenever we are working with the exponents, taking the appropriate radical will always cancel out the exponent operation. That is to say, if a number is raised to the power of a, we can cancel out the exponent by taking the resulting value and taking the a-th root of the result. This is the same as raising the the resulting number to a fractional exponent. For example, if we have 74, in order to cancel out the exponent 4 we would have to take the 4th root of 74 which is the same as raising 74 to the ¼ power. This is shown below:

fourth root of 7 space equals space left parenthesis space space 7 to the power of 4 space right parenthesis to the power of space 1 fourth end exponent space equals space 7 to the power of 4 over 4 end exponent equals space 7

In general, whenever we have an expression raised to the a-th power, we can take the a-th root of that number, which is the same as raising that number to the 1/a power. By doing this, we effectively cancel out the radical with the exponent.

a-th root of x to the power of a end root space equals space left parenthesis x to the power of a right parenthesis to the power of 1 over a end exponent equals x to the power of a over a end exponent equals x to the power of 1 space equals space x

Let’s look at this a little more in-depth. When we simplify radicals with exponents, we divide the exponent by the index. Another way to write division is with a fraction bar. This is how we will define rational exponents.

formula
Property of Fractional Exponents
a to the power of n over m end exponent space equals space left parenthesis m-th root of a right parenthesis to the power of n

The denominator of a rational exponent becomes the index on our radical. Likewise, the index on the radical becomes the denominator of the exponent. We can use this property to change any radical expression into an exponential expression.

open parentheses fifth root of x close parentheses cubed equals x to the power of 3 over 5 end exponent open parentheses root index 6 of 3 x end root close parentheses to the power of 5 equals open parentheses 3 x close parentheses to the power of 5 over 6 end exponent

Index is denominator
1 over open parentheses root index 7 of a close parentheses cubed equals a to the power of short dash 3 over 7 end exponent 1 over open parentheses cube root of x y end root close parentheses squared equals open parentheses x y close parentheses to the power of short dash 2 over 3 end exponent Negative exponents from reciprocals

We can also change any rational exponent into a radical expression by using the denominator as the index.

a to the power of 5 over 3 end exponent equals open parentheses cube root of a close parentheses to the power of 5 open parentheses 2 m n close parentheses to the power of 2 over 7 end exponent equals open parentheses root index 7 of 2 m n end root close parentheses squared

Index is denominator
x to the power of short dash 4 over 5 end exponent equals 1 over open parentheses fifth root of x close parentheses to the power of 4 open parentheses x y close parentheses to the power of short dash 2 over 9 end exponent equals 1 over open parentheses root index 9 of x y end root close parentheses squared Negative exponent means reciprocals

The ability to change between exponential expressions and radical expressions allows us to evaluate problems we had no means of evaluating before. We do this by changing the expression to a radical.

27 to the power of short dash 4 over 3 end exponent

Change to radical, denominator is index, negative means reciprocal
1 over open parentheses cube root of 27 close parentheses to the power of 4

Evaluate radical

summary
Exponents and radicals are inverse operations of each other, meaning they cancel each other out. Fractional exponents and radicals can be written as one another by using the property of fractional exponents. The largest advantage of being able to change a radical expression into an exponential expression is that we are now allowed to use all of our exponent properties to simplify.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html

Formulas to Know
Property of Fractional Exponents

a to the power of n over m end exponent equals m-th root of a to the power of n end root

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