Answer:
Rational Exponents
Learning Objective(s)
· Convert radicals to expressions with rational exponents.
· Convert expressions with rational exponents to their radical equivalent.
· Use the laws of exponents to simplify expressions with rational exponents.
· Use rational exponents to simplify radical expressions.
Introduction
Square roots are most often written using a radical sign, like this, . But there is another way to represent the taking of a root. You can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example,  can be written as .
Can’t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems. Let’s explore the relationship between rational (fractional) exponents and radicals.
Rewriting Radical Expressions Using Rational Exponents
Radicals and fractional exponents are alternate ways of expressing the same thing. You have already seen how square roots can be expressed as an exponent to the power of one-half.
Radical Form
Exponent Form
Integer


4


5


10
Let’s look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in these examples, the denominator of the rational exponent is the number 3.
Radical Form
Exponent Form
Integer


2


5


10
These examples help us model a relationship between radicals and rational exponents: namely, that the nth root of a number can be written as either  or .
Radical Form
Exponent Form






…
…


When faced with an expression containing a rational exponent, you can rewrite it using a radical. In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of  translates to the square root, an exponent of  translates to the fifth root or , and  translates to the eighth root or .
Example
Problem
Write  as an expression with a rational exponent.

The radical form  can be rewritten as the exponent . Remove the radical and place the exponent next to the base.
Answer

Example
Problem
Express  in radical form.

Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.
The parentheses in  indicate that the exponent refers to everything within the parentheses.
Answer

Remember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference—there are no parentheses! Look what happens.
