Homework Help: Simplifying Radicals

How to Simplify Radicals

A radical is the root (such as square root or cube root) of a number. The radicand is the number under the radical sign. Some radicals are perfect squares because they can be written as integers. For instance, radical nine is equal to three, since 3 x 3 = 9. Other perfect squares include radical 16, radical 25 and radical 36.

Some radicals are imperfect squares, because they are equal to decimals, not integers. For example, radical seven is equal to a decimal, so it is an imperfect square. Radical eight, radical 15 and radical 23 are all imperfect squares as well.

Simplifying Perfect Squares

If a radical is a perfect square, all you need to do to simplify it is to write it as the corresponding integer. For instance, you can simplify radical 16 by writing it as four. If a radical is an imperfect square, though, you need to take a different approach.

Simplifying Imperfect Squares

Radicals have factors just like integers do. You probably already know that two and three are factors of six, since 2 x 3 = 6, and one and six are also factors of six. The same idea applies to radicals. The factors of radical six include radical two, radical three, radical one and radical six.

Using Factors to Simplify Radicals

How does factoring radicals help us to simplify them? It works like this: if a radical has a perfect square as a factor, you can pull out that factor and simplify it to the corresponding integer. Here's an example:

(radical 27)

= (radical 9)(radical 3)

= 3(radical 3)

Since 3 x 9 = 27, that means that (radical 27) = (radical 9)(radical 3). Radical nine is a perfect square, so it can be simplified to three. Radical three is an imperfect square, so it cannot be simplified further. Here's another example that's a bit more complex:

(radical 80)

= (radical 4)(radical 20)

= (radical 4)(radical 4)(radical 5)

= (2)(2)(radical 5)

= 4(radical 5)

In this problem, radical 80 has radical four as a perfect-square factor, so we simplified it to (radical 4)(radical 20). Then, radical 20 was further simplified to (radical 4)(radical 5) to produce another perfect-square factor. Since both radical fours are equal to two, we reduced them and worked out the multiplication. The radical five can't be simplified any further, so we left it in our answer.

Tip: If your answer contains a radical that has a perfect square as a factor, it is not completely simplified.

Practice Problems

Simplify the following radicals. In some cases, the radicals may be perfect squares, and in other cases, it may not be possible to simplify them further.

1. Simplify radical 32.

2. Simplify radical 26.

3. Simplify radical 144.

Solutions

Keep in mind that since some radicals have multiple perfect-square factors, there may be different ways to arrive at the correct solution. If your answer matches the ones below, then it is correct even if you arrived at that answer in a different way.

1. The answer is 4(radical 2). Here's how to find it:

(radical 32)

= (radical 4)(radical 8)

= (radical 4)(radical 4)(radical 2)

= (2)(2)(radical 2)

= 4(radical 2)

2. Since radical 26 does not have any factors that are perfect squares, it cannot be simplified.

3. You may have known right away that radical 144 is a perfect square that's equal to 12. If not, you may have used this method to arrive at the same answer:

(radical 144)

= (radical 4)(radical 36)

= (2)(6) = 12

Here's another approach that you may have taken:

(radical 144)

= (radical 2)(radical 72)

= (radical 2)(radical 9)(radical 8)

= (radical 2)(radical 9)(radical 4)(radical 2)

= (3)(2)(radical 2)(radical 2)

= 6(radical 2)(radical 2)

You may have gotten stuck at this point, since radical two can't be simplified. How can you get 12 out of this? Since (radical 2)(radical 2) = (radical 4), you can proceed as follows:

= 6(radical 4)

= (6)(2)

= 12