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Exponents

The Square Root

The square root is the reverse operation of squaring a number.

Example 1.

3² = 9,  so 3 is a square root of 9.

(-3)² = 9,  so -3 is also a square root of 9.

Example

Since r² = -1 has no real number solution, -1 does not have a real square root.

• More generally, since b² ≥ 0 for any real number b, negative numbers have no real square root.
• Any positive number has two square roots.
• The number 0 has only one square root; namely 0.

For any positive number b, the positive square root is denoted by and the negative square root is denoted by .

The graph of x² is shown below. You can see that for any positive number b, there exists a positive and a negative root.

When writing the two square roots of a positive number b, we combine the two roots as .

For example, the square roots of 4 are and the square roots of 1 are .

Not all square roots of a number are integers. For example, and are not integers.

Exercise

If the square root of b exists and is an integer write it in the box then check it. If the square root exists but is not an integer, click on "Real Root is Not Integer". If the number b does not have a real square root, click on "No Real Root".