Square Root

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Square Root

The first step in learning about square roots is understanding the symbol for square roots. The symbol can be called a radical, square root, or root. There are three names for the same thing.

To indicate the square root of a number is needed to be solved, a number is inserted inside the square root symbol. The number inside the radical is also called the radicand. The following indicates the square root of 4 needs to be solved.

How to find the square root of a number

When a number is squared, it is raised to the 2nd power and the answer is its square. The square root solves for the root of the square, which is the original number that was raised to the 2nd power.

\begin{array}{c} {2^2} = 4\\ \sqrt 4 = 2\\ {3^2} = 9\\ \sqrt 9 = 3\\ {4^2} = 16\\ \sqrt {16} = 4 \end{array}

Another way to complete a square root is by raising it to the half power.

\begin{array}{c} {25^{1/2}} = 5\\ \sqrt {25} = 5 \end{array}

Perfect squares

Perfect squares are squares that have an integer for a square root. These are the simplest to solve since additional significant figures are needed. See the following perfect squares.

Multiplying square roots

When square roots are multiplied, they can be simplified into one term under the radical. This is also called the “product property of square roots”.

\begin{array}{l} \sqrt {ab} = \sqrt a *\sqrt b \\ \sqrt a *\sqrt b = \sqrt {ab} \end{array}

Here is one example of working through multiplying square roots.

\begin{array}{l} \sqrt 4 *\sqrt 9 \\ \sqrt {4*9} \\ \sqrt {36} = 6 \end{array}

How to simplify square roots

When a perfect square is not apparent in the radial, the number can be simplified by finding perfect squares within that number. This can be accomplished by breaking down the number into its smallest forms and combining similar terms to create a perfect square.

\begin{array}{c} \sqrt {54} \\ \sqrt {9*6} \\ \sqrt 9 *\sqrt 6 \\ 3*\sqrt 6 \end{array}

If a number under a radical can only be broken down into prime numbers, the number cannot be simplified. In the example below, 5 and 11 are not perfect squares so the square root of 55 cannot be simplified.

\begin{array}{l} \sqrt {55} \\ \sqrt {5*11} \\ \sqrt 5 *\sqrt {11} \end{array}

Adding square roots

To add square roots, the radicand must be like terms. If the radicand values are not like terms, they must be simplified to be solved.

\sqrt 3 + \sqrt 3 + \sqrt 3 = 3\sqrt 3

Perfect squares can be solved and added together.

\begin{array}{l} \sqrt 4 + \sqrt {25} + \sqrt 9 \\ 2 + 5 + 3 = 10 \end{array}

Frequently Asked Questions

What is the square root of 4?

The square root of 4 is 2.

What is the square root of 64?

The square root of 64 is 8.

More information

Review the exponents page for more information on how exponents work.

Check out the Algebra Index for more Algebra topics!

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