Simplifying Square Roots
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:
 Factor the number inside the
square root sign.
 If a factor appears twice, cross out both and write the factor one time to
the left of the square root sign. If the factor appears three times, cross out
two of the factors and write the factor outside the sign, and leave the third
factor inside the sign. Note: If a factor appears 4, 6, 8, etc. times, this
counts as 2, 3, and 4 pairs, respectively.
 Multiply the numbers outside the sign. Multiply the numbers left inside
the sign.
 Check: The outside number squared times the inside number should equal the
original number inside the square root.
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}.

=

= 2×

2× = 2×
 Check: 2^{2}×3 = 12
Example 2: Simplify
.

=

= 2×5×

2×5× = 10×
 Check: 10^{2}×6 = 600
Example 3: Simplify
.

=

= 3×3×

3×3× = 9×
 Check: 9^{2}×10 = 810
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.
Approximating Square Roots
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:
 Pick a perfect square that is close to the given number. Take its square
root.
 Divide the original number by this result.
 Take the arithmetic mean of the result of I and the result of II by
adding the two numbers and dividing by 2 (this is also called "taking an
average").
 Divide the original number by the result of III.
 Take the arithmetic mean of the result of III and the result of IV.
 Repeat steps IVVI using this new result, until the approximation is
sufficiently close.
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.