Square Roots and Cube 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 square root is equation. Following is a list of the first eleven perfect (whole number) square roots.

equation

Special note: If no sign (or a positive sign) is placed in front of the square root, then the positive answer is required. Only if a negative sign is in front of the square root is the negative answer required. This notation is used in many texts and is adhered to in this book. Therefore,

equation

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 equation. Other roots are defined similarly and identified by the index given. (In square root, an index of two is understood and usually not written.) Following is a list of the first eleven perfect (whole number) cube roots.

equation

Approximating square roots

To find the square root of a number that is not a perfect square, it will be necessary to find an approximate answer by using the procedure given in Example

.
Example 1

Approximate equation.

Since 62 = 36 and 72 = 49, then equation is between equation and equation.

Therefore, equation is a value between 6 and 7. Since 42 is about halfway between 36 and 49, you can expect that equation will be close to halfway between 6 and 7, or about 6.5. To check this estimation, 6.5 × 6.5 = 42.25, or about 42.

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:

equation

Simplifying square roots

Sometimes you will have to simplify square roots, or write them in simplest form. In fractions, equation can be reduced to equation. In square roots, equation can be simplified to equation.

There are two main methods to simplify a square root.

Method 1: Factor the number under the equation into two factors, one of which is the largest possible perfect square. (Perfect squares are 1, 4, 9, 16, 25, 36, 49, …)

Method 2: Completely factor the number under the equation into prime factors and then simplify by bringing out any factors that came in pairs.

Example 2

Simplify equation.

equation

In Example

, the largest perfect square is easy to see, and Method 1 probably is a faster method.
Example 3

Simplify equation.

equation

In Example

, it is not so obvious that the largest perfect square is 144, so Method 2 is probably the faster method.

Many square roots cannot be simplified because they are already in simplest form, such as equation, equation, and equation.