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:

1. Factor the number inside the square root sign.
2. 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.
3. Multiply the numbers outside the sign. Multiply the numbers left inside the sign.
4. 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.

1. =
2. = 2×
3. 2× = 2×
4. Check: 2²×3 = 12

Example 2: Simplify .

1. =
2. = 2×5×
3. 2×5× = 10×
4. Check: 10²×6 = 600

Example 3: Simplify .

1. =
   
2. = 3×3×
   
3. 3×3× = 9×
4. Check: 9²×10 = 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.

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:

1. Pick a perfect square that is close to the given number. Take its square root.
2. Divide the original number by this result.
3. 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").
4. Divide the original number by the result of III.
5. Take the arithmetic mean of the result of III and the result of IV.
6. Repeat steps IV-VI 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.

Here are some examples to make the steps clearer:

Example 1: Approximate .

1. 25 is close to 22. = 5
2. 22/5 = 4.4
3. (5 + 4.4)/2 = 4.7
4. 22/4.7 = 4.68
5. (4.7 + 4.68)/2 = 4.69
6. 22/4.69 = 4.69

= 4.69

Example 2: Approximate .

1. 71 is close to 64. = 8
2. 71/8 = 8.9
3. (8 + 8.9)/2 = 8.45
4. 71/8.45 = 8.40
5. (8.45 + 8.40)/2 = 8.425
6. 71/8.425 = 8.427
7. (8.425 + 8.427)/2 = 8.426
8. 71/8.426 = 8.426

= 8.426

Example 3: Approximate .
can be simplified: = = 2× = 2×
Approximate :

1. 14 is close to 16. = 4
2. 14/4 = 3.5
3. (4 + 3.5)/2 = 3.75
4. 14/3.75 = 3.73
5. (3.75 + 3.73)/2 = 3.74
6. 14/3.74 = 3.74

= 3.74

Thus, = 2× = 2×3.74 = 7.48

Note that the eventual result will be the same no matter what perfect square one picks in Step 1.