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Roots and Square Roots

Since a root is the number of times a number can be multiplied by itself to go into another number, using roots (and square roots) is quite basic. Also, if you have already gone through the Exponents tutorials, then you already know that a root is the inverse of an exponent. Or, if you wanted, you could write a root as an exponent.

A "square root" is a number, when multipled by itself two times, gives you the other number. For example, the square root of 9 is 3, since 3x3=9.
A "cubed root" is a number, when multipled by itself three times, gives you the other number. For example, the cube root of 8 is 2, since 2x2x2=8.
To find larger roots, continue the pattern of multiplication. For example, the seventh root of a 128 is 2, since 2x2x2x2x2x2x2=128.

Let's try a couple:
1. Find the square root of 25.
2. Find the square root of 20.

1. This is a perfect square, meaning there will be no decimals in the answer. We can just start using small numbers until we find one that, when multipled by itself, equals 25. For the sake of showing how easy this is, we will start with 1.
1x1 = 1. Not the answer.
2x2=4. Not the answer.
3x3=9. Not the answer.
4x4=16. Not the answer. 5x5=25. The Answer!
6x6=36. Not the answer; 36 > 25, so it must be less than 6.

2.. To find a square root, follow these steps:

   a. Guess a number you think is close. To determine a "close" number, just multiple a couple numbers in your head. For example, if you are attempting to find the square root of 20, you know that 4x4=16 and 5x5=25. So the square root is somewhere between 4 and 5. We'll go with 4, since 4 squared is closer to 20 than 5 squared.

   b. Divide the original number by your guess. So using 20, we divide it by our guess, 4. That gives us an answer of 5.

   c. Average your guess and the divided value together. (Add them and divide by 2.) Our guess of 4 + our divided answer of 5 = 9. Nine divided by 2 = 4.5.

   d. Repeat steps (b) and (c) until you've arrived at a close enough approximation. (Using a calculator, the answer is 4.4721, which depending on the type of work you are doing, is very close.


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