Math Help: Understanding Square Roots

Help with Square Roots

A square root is a number that, when multiplied by itself, equals another number. For example, the square root of 25 is five, because 5 x 5 = 25. Here are some other examples of square roots:

• Square root of 4 = 2
• Square root of 9 = 3
• Square root of 16 = 4
• Square root of 36 = 6

Irrational Square Roots

The square roots of some numbers are impossible to define, because they are equal to decimals that continue infinitely in an unpredictable pattern. This means that they can't be written as fractions, so they are irrational numbers. Examples of this include the square root of two and the square root of three.

Estimating the Values of Square Roots

Some numbers have decimals instead of integers as square roots. For instance, the square root of seven is approximately 2.646. Teachers may ask you to estimate the value of square roots like these, or find their approximate locations on a number line without using a calculator.

As an example, let's estimate the value of the square root of two:

• Square root of 1 < square root of 2 < square root of 4
• Square root of 1 = 1
• Square root of 4 = 2

The value of the square root of two must be greater than one and less than two.

Operations with Square Roots

Square roots and terms that contain square roots can be added, subtracted and multiplied if certain rules are followed. These rules are similar to the rules for operations with terms containing unknown variables, since the exact value of a square root is often unknown in an equation.

Adding and Subtracting Square Roots

Square roots and terms containing square roots can be added or subtracted if the square roots are the same. For instance, (square root of 7) + (square root of 5) cannot be simplified further; however, (square root of 7) + (square root of 7) = 2(square root of 7). This rule applies to terms with square roots as well. You cannot simplify 8(square root of 7) - 2(square root of 4) any further, but 8(square root of 7) - 2(square root of 7) = 6(square root of 7).

Multiplying Square Roots

You can multiply square roots together just like you multiply integers. For instance, (square root of 4) x (square root of 5) = (square root of 20).

Practice Problems

1. Seven is the square root of what number?

2. Find the square root of 64.

3. Is the square root of 14 greater than or less than four?

4. Simplify this expression as much as possible: 2(square root of 3) + 4(square root of 3) - (square root of 7).

5. Simplify the following expression: (square root of 7) x (square root of 7).

Solutions

1. Since 7 x 7 = 49, seven is the square root of 49.

2. The square root of 64 is eight, because 8 x 8 = 64.

3. The square root of 14 is less than four. You know that (square root of 14) < (square root of 16), and (square root of 16) = 4, so (square root of 14) < 4.

4. Since 2(square root of 3) and 4(square root of 3) have the same square root, they can be added together to get 6(square root of 3). The square root of seven cannot be subtracted because it is different from the square root of three, so the final answer is 6(square root of 3) - (square root of 7).

5. The product of the square root of seven and itself is the square root of 49, since 7 x 7 = 49. Since (square root of 49) = 7, seven is the final answer.