nth Roots and Rational Exponents - How to Simplify

How to Simplify nth Roots with Rational Exponents - Video

Root Vocabulary

Root Vocabulary:

First we have n√a which is the nth root of a. The n is the index, √ is the radical sign, and the a is the radicand. And a1/2 is the root. With numbers we have 2√49 which is the square root of 49. And 7 is the root.

nth Roots:

Here we have what happens when n, the index, is even or odd.

When n is even, we have 3 possibilities when a is negative, 0, or positive. If a is less than 0, then there are not any real nth roots. If a is 0, then there is only one nth root, 0. And if a is greater than 0, there are two real nth roots, ±a1/n.

When n is odd, we have 3 possibilities when a is negative, 0, or positive. If a is less than 0, then there is one real nth root, a1/n. If a is 0, then there is only one nth root, 0. And if a is greater than 0, there is one real nth root, a1/n.

Rational Exponents:

Here we have what happens when n, the index, is already an exponent.

One method is take the numerator of the exponent outside the base and convert the denominator to the index of the radical sign. And if the exponent is negative just move the base to from the numerator to the denominator or vice versa and switch the sign of the exponent.

Example 1 

Example 1:

We have the expression 3√24. Our first step is find the prime factorization of 24, and we get 2 * 2 * 2 * 3. We have two ways to approach this problem. One use the rational exponents or the radical sign.

Let's start off with the rational exponents. And continue with the radical sign.

Example 1 continued:

Here we have two ways to get the same answer. With the rational exponents, there are a few more steps but they show why the index and the exponent cancel out so we are left with just the number underneath the radical sign. And in this case, it is 2(31/3) or 2 3√3.

Example 2

Example 2:

We have the expression (4√16)-2. Our first step is find the prime factorization of 16, and we get 2 * 2 * 2 * 2. We have two ways to approach this problem. One use the rational exponents or the radical sign.

Let's start off with the rational exponents. And continue with the radical sign.

Example 2 continued:

Here we have two ways to get the same answer. With the rational exponents, there are a few more steps but they show why the index and the exponent cancel out so we are left with just the number underneath the radical sign. And in this case, it is 1/4 or 1/4.

Live Worksheet

Teacher - Edpuzzle Link