# Square root

A square root is a number that can be multiplied by itself to give us the value under the radical symbol (also referred to as the radix). A square root is denoted as follows:

In the figure above, z is the square root of x. We can also write this as "z raised to the power of 2 is equal to x" as follows:

z^{2} = x

Recognizing perfect squares is useful for solving equations as well as simplifying radical expressions. One important thing to note is that the square root actually has two solutions, a positive and a negative one. This is because a negative number multiplied by itself is a positive number (and a positive number multiplied by itself is also a positive number). Therefore, the square root of 9 is either -3 or 3, since both 3 × 3 and -3 × -3 equal 9. This is particularly important to remember when solving algebraic equations involving square roots. However, when written with the radical symbol, we usually just give the positive solution.

Examples

1. Simplify:

Using one of the properties of radicals:

In this case, knowing that 4 is a perfect square allowed us to simplify the radical.

2. Solve: 2x^{2} - 8 = 0

The fact that x must equal both positive and negative 2 is clearer if we solve the above equation by factoring:

We solve this by dividing both sides by either (x + 2) or (x - 2), then solving the equation using the remaining term. If we solve x + 2 = 0 first, we get x = -2. Solving for x - 2 = 0 gives us 2, showing us that there are two solutions.

## Square roots of non-perfect squares

Finding the square root of a perfect square is relatively simple. As long as you recognize some perfect squares, you can probably square a few numbers that are close and find the solution fairly quickly.

However, calculating the square root of non-perfect squares is much more difficult, and is usually not something we are required to do. If we have to, we can estimate a square root by guessing values between those of known perfect squares.

Example

1. Estimate :

We know that the square root of 4 is 2, and that the square root of 1 is 1. So we know that the square root of 2 must be between 1 and 2. Since 2 is closer to 1 than it is to 4 (we skipped 3 because it is also not a perfect square), we can guess that the value of its square root is closer to that of the square root of 1, than it is to the square root of 4, so it should be less than 1.5.

From there, we could keep guessing values and manually multiplying them by themselves to get closer and closer to 2, but this is very tedious. The square root of 2 is an irrational number, the first digits of which are 1.41421.

There are also algorithms for calculating the square root, but it is unlikely that you will need to calculate a square root with precision by hand, so for most cases where a calculator is not available, an approximation should suffice.