Finding Squares and Square Roots Study Guide: GED Math

Practice problems for these concepts can be found at:

Exponents and Roots Practice Problems: GED Math

RECALL THAT A FACTOR of a number is a whole number that divides evenly, without a remainder, into the given number. Frequently, you multiply the same factor times itself several times. In math, there is a special notation for this idea: exponents and the inverse operation, roots. This lesson explores square roots.

Finding Squares

Anytime a number is written with a 2 raised after it, it means to multiply the number by itself, or to square the number. A square of a number is just the number multiplied by itself. For example, 4² is called "four squared." Because 4² = 4 × 4, which is 16, 16 is called a perfect square.

Examples

1. What is the square of 30?

To find the square of a number, multiply it by itself. The square of 30 is 30 × 30, or 900.

2. Find 9².

When a number is followed by a raised 2, you should square it: 9² = 9 × 9 = 81.

Although you can always calculate the square of a number by multiplying, it's a good idea to know some of the perfect squares, both for raising to an exponent and for taking roots (discussed on page 80).

Here are some common squares you might want to learn.

Square Roots

You may have seen this symbol before: . This is the symbol for a square root. When a number is written after it, you are being asked to find the square root of that number. In an expression such as , 25 is called the radicand, and the expression is the radical.

The square root of a number is one of the two identical factors whose product is the given number. For example, 64 is a perfect square because 8 × 8, or 8², equals 64. This factor, 8, is called the square root of 64.

The number 64 has another square root, –8, because –8 × –8 = (–8)² = 64. If we want to indicate the positive square root, use the radical symbol to denote square root. So = 8, and – = –8.

Working with Square Roots

You can simplify square roots by expressing the radicand (the number after the radical symbol) as the product of other numbers, where one of the factors is a perfect square. For example, , which means 12 times the square root of 2.

Likewise, when you are given a problem of two radicals multiplied together, you can combine by multiplying the radicands: .

The same rule holds for division: and .

You can add or subtract radicals only if they have the same radicand. 5 + 7 = 12 , and 7 – 10 = –3 , but 6 + 7 cannot be combined because the radicands are different.

Examples

1. What is ?

The problem is asking you to calculate the square root of 25. Ask yourself what number multiplied by itself equals 25. If you have memorized the list of common squares, this problem is not very hard. Even if you haven't learned the list of common squares yet, though, you can figure this problem out: 5 × 5 = 25. So the square root of 25 is 5.

2. What is ?

The problem is asking you what number equals 45 when multiplied by itself. You know that 62 = 36 and 7² = 49. Thus, the square root of 45 is a number between 6 and 7. You can find a more precise answer using a calculator.

Practice problems for these concepts can be found at:

Exponents and Roots Practice Problems: GED Math