nth Roots - How to Simplify Multiple Terms just Numbers

How to Simplify Multiple nth Root Terms Just with Numbers - 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 5√85√4. Our first step is find the prime factorization of 8, and we get 2 * 2 * 2. Next we need to find the prime factorization of 4, and we get 2 * 2.

Now that we have the prime factorization of each number, we can rearrange 8 and 4 so that we can simplify a little easier using the index. 

Example 1 continued:

Here we have shown what would happen if you multiplied the radical expressions. In this case, the numbers are small enough so that it wouldn't matter, but in other cases the numbers might become really big so it is worth getting the numbers small.

Let's go over the steps if we did multiply.

So the answer comes out the same. Luckily for us, in this case the radicand is still quite small so the amount steps to solve is about the same.

Example 2

Example 2:

We have the expression 4√343(34√7). Our first step is find the prime factorization of 343, and we get 7 * 7 * 7. Next we need to find the prime factorization of 7, and luckily 7 is already a prime so we get 7.

Now that we have the prime factorization of each number, we can rearrange 343 and 7 so that we can simplify a little easier using the index. 

So the expression 4√343(34√7) is 21. This time, we are not going to multiply 343 and 7 because that would 2401. That defeats the purpose of simplify because now we have a big number.

Example 3

Example 3:

We have the expression -3√4 / 3√5. Our first step is find the prime factorization of 4, and we get 2 * 2 . Next we need to find the prime factorization of 5, and luckily 5 is already a prime so we get 5.

Now that we have the prime factorization of each number, we can rearrange 4 and 5 so that we can simplify a little easier using the index. 

Since we have a radical in the denominator, we have to rationalize it. It makes it look nicer and it is easier to divide a whole number than a radical if we want to divide.

In order to rationalize, we look at the index. Since ours is 3, we need our power to be 3. We already have 1 in the denominator so we need 2 more. Now we need to multiply the numerator and denominator by 3√52. So we can get 3√53 get in the denominator so we can simplify. After we multiply the numerator by 3√52, we get 3√100 numerator. So the final answer is -3√100 / 5.

Live Worksheet

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