## Square Roots

A*square root*is the factor of a number that, when multiplied by itself, produces the number. Take the number 36, for example. One of the factors of 36 is 6. If you multiply 6 by itself (6 × 6), you come up with 36, so 6 is the square root of 36. The number 36 has other factors, such as 18. But if you multiply 18 by itself (18 × 18), you get 324, not 36. So, 18 isn’t the square root of 36.

All whole numbers are grouped into one of two camps when it comes to roots:

**Perfect squares:**Only a few whole numbers, called*perfect squares,*have exact square roots. For example, the square root of 25 is 5.**Irrational numbers:**Other whole numbers have square roots that are decimals that go on forever and have no pattern that repeats (nonrepeating, nonterminating decimals), so they’re called*irrational numbers.*The square root of 30 is 5.4772255 with no end to the decimal places, so the square root of 30 is an irrational number.

*radical sign*. It looks like this:

.Here’s how you use it:

means “the square root of 36” — in other words, 6.

## Perfect squares

Square roots can be difficult to find at times without a calculator, but because you can’t use a calculator during the test, you’re going to have to use your mind and some guessing methods. To find the square root of a number without a calculator, make an educated guess and then verify your results.To use the educated-guess method (see the next section), you have to know the square roots of a few perfect squares. One good way to do this is to memorize the squares of the square roots 1 through 12:

- 1 is the square root of 1 (1 × 1 = 1)
- 2 is the square root of 4 (2 × 2 = 4)
- 3 is the square root of 9 (3 × 3 = 9)
- 4 is the square root of 16 (4 × 4 = 16)
- 5 is the square root of 25 (5 × 5 = 25)
- 6 is the square root of 36 (6 × 6 = 36)
- 7 is the square root of 49 (7 × 7 = 49)
- 8 is the square root of 64 (8 × 8 = 64)
- 9 is the square root of 81 (9 × 9 = 81)
- 10 is the square root of 100 (10 × 10 = 100)
- 11 is the square root of 121 (11 × 11 = 121)
- 12 is the square root of 144 (12 × 12 = 144)

## Irrational numbers

When the ASVAB asks you to figure square roots of numbers that aren’t perfect squares, the task gets a bit more difficult. In this case, the ASVAB usually asks you to find the square root to the nearest tenth.Suppose you run across this problem:

Think about what you know: You know that the square root of 49 is 7, and 54 is slightly greater than 49. You also know that the square root of 64 is 8, and 54 is slightly less than 64. So, if the number 54 is somewhere between 49 and 64, the square root of 54 is somewhere between 7 and 8. Because 54 is closer to 49 than to 64, the square root will be closer to 7 than to 8, so you can try 7.3 as the square root of 54.

Multiply 7.3 by itself: 7.3 × 7.3 = 53.29, which is very close to 54. Now try multiplying 7.4 by itself to see if it’s any closer to 54: 7.4 × 7.4 = 54.76, which isn’t as close to 54 as 53.29. Therefore, 7.3 is the square root of 54 to the nearest tenth without going over.

## Other roots

The wonderful world of math is also home to concepts like cube roots, fourth roots, fifth roots, and so on. A*root*is a factor of a number that when cubed (multiplied by itself three times), taken to the fourth power (multiplied by itself four times), and so on produces the original number. A couple of examples seem to be in order:

- The cube root of 27 is 3. If you cube 3 (also known as raising it to the third power or multiplying 3 × 3 × 3), the product is 27.
- The fourth root of 16 is the number that, when multiplied by itself four times, equals 16. Any guesses? Drumroll, please: 2 is the fourth root of 16 because 2 × 2 × 2 × 2 = 16.