Square Roots and Cube Roots

Square roots

To find the square root of a number, you want to find some number that when multiplied by itself gives you the original number. In other words, to find the square root of 25, you want to find the number that when multiplied by itself gives you 25. The square root of 25, then, is 5. The symbol for the square root is . Following is a partial list of perfect (whole number) square roots.

Note: If no sign (or a positive sign) is placed in front of the square root, the positive answer is required. No sign means that a positive is understood. Only if a negative sign is in front of the square root is the negative answer required. Therefore,

Cube roots

To find the cube root of a number, you want to find some number that when multiplied by itself twice gives you the original number. In other words, to find the cube root of 8, you want to find the number that when multiplied by itself twice gives you 8. The cube root of 8, then, is 2 because 2 × 2 × 2 = 8. Notice that the symbol for cube root is the radical sign with a small three (called the index) above and to the left . Other roots are similarly defined and identified by the index given. (In square root, an index of 2 is understood and usually not written.) Following is a partial list of perfect (whole number) cube roots.

Approximating square roots

Example 1: Approximate .

Since 42 is almost halfway between 36 and 49, is almost halfway between is approximately 6.5. To check, multiply the following:

Example 2: Approximate

Since is slightly closer to than it is to ,

Check the answer.

Example 3: Approximate .

First, perform the operation under the radical.

Since is slightly closer to than it is to ,

Square roots of nonperfect squares can be approximated, looked up in tables, or found by using a calculator. You may want to keep these two in mind, because they are commonly used.

Simplifying square roots

Sometimes, you have to simplify square roots or write them in simplest form. In fractions, 2/4 can be reduced to ½. In square roots, can be simplified to . To simplify a square root, first factor the number under the into two factors, one of which is the largest possible perfect square. (Perfect square numbers are 1, 4, 9, 25, 49, and so on.)

Example 4: Simplify .

Then take the square root of the perfect square number.

Finally, write it as a single expression

Example 5: Simplify

To check, square the number on the outside of the radical and multiply it by the number on the inside.

Example 6: Simplify .

Remember: Most square roots cannot be simplified, because they are already in simplest form, such as .