Math Homework Help: Evaluating Radicals

How to Simplify Radicals

The radical sign, which looks like a check mark with a horizontal line extending from the top, has a number on the inside called the radicand. When the radical sign is used alone, you'll need to find the square root of the radicand. If a small 'three' is written next to the symbol, you'll find the radicand's third root (also known as the cube root). If a four, five, six or any other number appears beside the radical sign, you'll find the radicand's corresponding root.

Square Roots

A number's root can be multiplied by itself a certain number of times to produce the original number. The square root of a number equals that number when it's multiplied by itself once. For instance, (square root of 9) = 3 because 3 x 3 = 9. Likewise, (square root of 16) = 4 because 4 x 4 = 16. Here are a few more examples of square roots:

(square root of 49) = 7

(square root of 81) = 9

(square root of 144) = 12

Cube Roots

A number's cube root has to be multiplied by itself twice to produce the original number. For example, since 2 x 2 x 2 = 8, the (cube root of 8) = 2. Three is the cube root of 27 because 3 x 3 x 3 = 27. These are some other cube root examples:

(cube root of 64) = 4

(cube root of 125) = 5

(cube root of 1,000) = 10

Complex Radicals

You'll also encounter radicals that aren't perfect squares or cubes, like (square root of 28). To simplify a radical like this, you'll need to apply the following property:

(square root of ab) = (square root of a) x (square root of b)

This means you can break a radical into the product of two different radicals by factoring the radicand, like this:

(square root of 28) = (square root of 4) x (square root of 7)

Hopefully, you'll be able to find at least one factor of the radicand that's a perfect square, as we did in this problem. Since (square root of 4) = 2, we can simplify the expression to 2(square root of 7). The original radical is now in its simplest form because (square root of 7) can't be simplified any further. Here's an example with cube roots:

(cube root of 40) = (cube root of 8) x (cube root of 5) = 2(cube root of 5)