**Estimating Square Roots** *KS3 Revision*

## What you need to know

**Things to remember:**

- You should know your squared numbers from 1 – 144
- Think about the square numbers on either side of the number you are estimating the square root of.

Before we start, what does it mean to square or square root a number? Well, squaring is just multiplying a number by itself, like 3\times3=9 or 8\times8=64. Because you know your times tables up to 12\times12, you should also know your square numbers up to 12\times12. We use the special “power” or “index” notation of writing a 2 to the top right of a number, like 6^2=6\times6=36.

1^2=1\times1=1

2^2=2\times2=4

3^2=3\times3=9

4^2=4\times4=16

5^2=5\times5=25

6^2=6\times6=36

7^2=7\times7=49

8^2=8\times8=64

9^2=9\times9=81

10^2=10\times10=100

11^2=11\times11=121

12^2=12\times12=144

** **

And how about square rooting a number? This is just finding the number that we multiply by itself to make the number we started with. We have special notation for writing square roots, that look like a combination of a tick and the bus stop \sqrt{36}.

A typical square root question would look like this:

*What is \sqrt{36}?*

* *

If it helps you can think of this as “What number do we multiply by itself to make 36?” Well, we know we multiply 6 by itself to make 36, so the square root of 36 must be 6.

6\times6=36

\sqrt{36}=6

*What is \sqrt{81}?*

Let’s read it again as “What number do we multiply by itself to make 81?” Well, we know we multiply 9 by itself to make 81, so the square root of 81 must be 8.

9\times9=81

\sqrt{81}=9

*What is \sqrt{76}*

Let’s read it again as “What number do we multiply by itself to make 76?” Well, we know we multiply… Wait, 76 isn’t one of our square numbers. Hmmm, Let’s look at where 76 would fit into our list of square numbers.

8^2=8\times8=64

?^2=?\times?=76

9^2=9\times9=81

So, 76 is between 64 and 81, so our missing square root must be somewhere between 8 and 9. But, because this doesn’t give us an actual answer, it is just an estimate! So, really, this question should have been “Find an estimate for \sqrt{76}”

## Example Questions

**Question 1:** Find an estimate for \sqrt{40}

40 is between 36 and 49, so:

6^2=6\times6=36

?^2=?\times?=40

7^2=7\times7=49

An estimate for \sqrt{40} is that it is somewhere between 6 and 7.

**Question 2:** Find an estimate for \sqrt{135}

135 is between 121 and 144, so:

11^2=11\times11=121

?^2=?\times?=135

12^2=12\times12=144

An estimate for \sqrt{135} is that it is somewhere between 11 and 12.