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
Since 62 = 36 and 72 = 49, then is between and .
Therefore, is a value between 6 and 7. Since 42 is about halfway between 36 and 49, you can expect that 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:
Simplifying square roots
Sometimes you will have to simplify square roots, or write them in simplest form. In fractions, can be reduced to . In square roots, can be simplified to .
There are two main methods to simplify a square root.
Method 1: Factor the number under the 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 into prime factors and then simplify by bringing out any factors that came in pairs.
, the largest perfect square is easy to see, and Method 1 probably is a faster method.
, 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 , , and .