Science, Maths & Technology

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Squares, roots and powers

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# 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?

You probably spotted that 49 is 7 × 7, or 72, so she could make a square patio 7 slabs by 7 slabs.

Since 72 = 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.