Become familiar with roots for the ASVAB. A root is the opposite of a power or an exponent. There are infinite kinds of roots. You have the square root, which means “undoing” a base to the second power; the cube root, which means “undoing” a base raised to the third power; a fourth root, for numbers raised to the fourth power; and so on.
Square roots
A math operation requiring you to find a square root is designated by the radical symbol
The number underneath the radical line is called the radicand. For example, in the operation
the number 36 is the radicand.
A square root is a number that, when multiplied by itself, produces the radicand. Take the square root of 36
If you multiply 6 by itself (6 × 6), you come up with 36, so 6 is the square root of 36.
When you multiply two negative numbers together, you get a positive number. For example, –6 × –6 also equals 36, so –6 is also the square root of 36.
When you take a square root, the results include two square roots — one positive and one negative.
Computing the square roots of negative numbers, such as
is also possible, but it involves concepts such as imaginary numbers that you won’t be asked about.
Square roots come in two flavors:
Perfect squares: Only a few numbers, called perfect squares, have exact square roots.
Irrational numbers: All the rest of the numbers have square roots that include decimals that go on forever and have no pattern that repeat, so they’re called irrational numbers.
Perfect squares
Because you can’t use a calculator during the test, you’re going to have to use your mind and some guessing methods. Make an educated guess and then verify your results.
The radical symbol indicates that you’re to find the principal square root of the number under the radical. The principal square root is a positive number. But if you’re solving an equation such as x2 = 36, then you give both the positive and negative roots: 6 and –6.
To use the educated-guess method, you have to know the square roots of a few perfect squares. One good way to do so is to study the squares of the numbers 1 through 12:
1 and –1 are both square roots of 1.
2 and –2 are both square roots of 4.
3 and –3 are both square roots of 9.
4 and –4 are both square roots of 16.
5 and –5 are both square roots of 25.
6 and –6 are both square roots of 36.
7 and –7 are both square roots of 49.
8 and –8 are both square roots of 64.
9 and –9 are both square roots of 81.
10 and –10 are both square roots of 100.
11 and –11 are both square roots of 121.
12 and –12 are both square roots of 144.
Irrational numbers
If you have to find the square root of a number that isn’t a perfect square, 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:
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.
Try multiplying 7.4 by itself to see whether it’s any closer to 54.
7.4 × 7.4 = 54.76, which isn’t as close to 54 as 53.29.
So 7.3 is the square root of 54 to the nearest tenth.
Cube roots
A cube root is a number that when multiplied by itself three times equals the number under the radical. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. A cube root is expressed by the radical sign with a 3 written on the left of the radical.
You may see one or two cube-root problems on the math subtests of the ASVAB, but probably not more than that.
Unlike square roots, numbers only have one possible cube root. If the radicand is positive, the cube root will be a positive number.
Also, unlike square roots, finding the cube root of a negative number without involving advanced mathematics is possible. If the radicand is negative, the cube root will also be negative. For example,
Just like square roots, you should memorize a few common cube roots:
1 is the cube root of 1, and –1 is the cube root of –1.
2 is the cube root of 8, and –2 is the cube root of –8.
3 is the cube root of 27, and –3 is the cube root of –27.
4 is the cube root of 64, and –4 is the cube root of –64.
5 is the cube root of 125, and –5 is the cube root of –125.
6 is the cube root of 216, and –6 is the cube root of –216.
7 is the cube root of 343, and –7 is the cube root of –343.
8 is the cube root of 512, and –8 is the cube root of –512.
9 is the cube root of 729, and –9 is the cube root of –729.
10 is the cube root of 1,000, and –10 is the cube root of –1,000.