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
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,
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
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
Since 62 = 36 and 72 = 49, then
Therefore,
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:
Simplifying square roots
Sometimes you will have to simplify square roots, or write them in simplest form. In fractions,
There are two main methods to simplify a square root.
Method 1: Factor the number under the
Method 2: Completely factor the number under the
Example 2
Simplify
In Example
, the largest perfect square is easy to see, and Method 1 probably is a faster method.Example 3
Simplify
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