# 1.3 Square roots

Given any number, you now know how to find its square. But, given the squared number, how do you find the original number?

## Example 3

If the gardener in Example 1 had only 49 paving slabs, what size of square patio could she make?

### Answer

You probably spotted that 49 is 7 × 7, or 7^{2}, so she could make a square patio 7 slabs by 7 slabs.

Since 7^{2} = 49, 7 is the square root of 49, written

7 = .

Sometimes you can just look at a number and spot its square root, if the number is a ‘perfect square’ (i.e. the result of squaring a whole number). For example, 25 is a perfect square, and = 5. But more often than not you will need to use your calculator for square roots, and it is important to be able to find rough estimates as a check on your calculator work. So if you wanted , you would know that it would lie between = 7 and = 8, and you would expect an answer of seven point something. (It is 7.416 …).

Technically, ^{−}7 is also a square root of 49, since (^{−}7)^{2} = 49. This is called the negative square root. The sign √ is customarily used to denote the positive square root, so = 7 and ^{−} = ^{−}7.

In Example 3, only the positive square root is relevant (patios have positive length sides).

## Example 4

The owners of a new house, with a bare earth garden, see an advertisement for 44 square metres of turf, ‘free to a good home – only pay transportation’. They were planning a square lawn surrounded by flower beds.

Find a rough estimate for the square root of 44. Use your calculator to find and find the size of the square lawn which the turf would make.

### Answer

Rounding 44 down to 40 doesn't help – you don't know the square root of 40 either! But you do know that 6^{2} = 36 (which is less than 44) and 7^{2} = 49 (which is greater than 44) so lies between 6 and 7. You could leave the answer as ‘between 6 and 7’ or guess it as 6.5, say. The calculated answer is 6.6332 (rounded to four decimal places). So the turf would make a lawn about 6.6 m square.