C. Rational Exponent Laws

C. Rational Exponent Laws

So far in Lesson 4.4, you have observed powers containing Whole Number exponents, zero exponents, and negative exponents. What happens when the exponent of a power is a fraction? The Power of a Power Law of Exponents can help define a rule for powers with rational exponents and illustrate their connection to radicals.

Rational Exponent an exponent that is in the form of a fraction,

Recall that for positive values of a, the square root of a² is a. In other words, taking the square root is the opposite operation of squaring.

This is equivalent to .

Notice that an equation equivalent to a¹ can be derived using exponential expressions and the Power of a Power Law.

Looking at the exponents on left side of the equation, what value, when multiplied by 2, results in the exponent of 1 found on the right side of the equation?

A value of will work.

You can confirm by applying the Power of a Power Law and simplifying.

The left side of the equation is equal to the right side.

The result of these findings can be summarized as follows:

In other words, taking a value to the exponent of is the same as taking the square root of that value. This statement can be generalized for any exponent of the form . Therefore, taking a value to the exponent is the same as taking the n^th root: .