Often, it becomes necessary to simplify a square root; that is, to remove all factors that are perfect squares from inside the square root sign and place their square roots outside the sign. This action ensures that the irrational number is the smallest number possible, making it is easier to work with. To simplify a square root, follow these steps:
To simplify the square root of a fraction, simplify the numerator and simplify the denominator.
Here are some examples to make the steps clearer:
Example 1: Simplify
12^{1/2}
.
Similarly, to simplify a cube root, factor the number inside the " ( )^{1/3} " sign. If a factor appears three times, cross out all three and write the factor one time outside the cube root sign.
It is very difficult to know the square root of a number (other than a perfect square) just by looking at it. And one cannot simply divide by some given number every time to find a square root. Thus, is it helpful to have a method for approximating square roots. To employ this method, it is useful to first memorize the square roots of the perfect squares. Here are the steps to approximate a square root:
If the square root can be simplified, it is easier to simplify and then approximate the number inside the " ( )^{1/2} " sign. This result can then be multiplied by the number outside the " ( )^{1/2} " sign.
Here are some examples to make the steps clearer:
Example 1: Approximate
.
Note that the eventual result will be the same no matter what perfect square one picks in Step 1.