 Relationship Between Squares and Square Roots | KS3 Maths | MME

## What you need to know

Things to remember:

• If we square a number and then square root it, we get the number we squared.
• If we square root a number and then square it, we get the number we square rooted.
• Because you should know your times tables up to $12\times12$, you should know all your square numbers up to 144.

A square number is a number that we find by square another number by itself.

$$4^2=4\times4=16$$

$$7\times7=49$$

$$11\times11=121$$

Square rooting tells us which number we multiply by itself to find our original number.

$$\sqrt{16}=4$$

$$\sqrt{49}=7$$

$$\sqrt{121}=11$$

We’re going to look at two mathematical journeys:

• Start with a number, square it, square root it, and see what we get.
• Start with a number, square root it, square it, and see what we get.

$$5\rightarrow5^2=25\rightarrow\sqrt{25}=5\rightarrow5$$

$$9\rightarrow9^2=81\rightarrow\sqrt{81}=9\rightarrow9$$

$$12\rightarrow12^2=144\rightarrow\sqrt{144}=12\rightarrow12$$

So, when we square a number and then square root it, it gives us the number we started with. How about if we square rooted first?

$$49\rightarrow\sqrt{49}=7\rightarrow7^2=49\rightarrow49$$

$$64\rightarrow\sqrt{64}=8\rightarrow8^2=64\rightarrow64$$

$$36\rightarrow\sqrt{36}=6\rightarrow6^2=36\rightarrow36$$

So, when we square root a number and then square, it gives us the number we started with.

Putting these two points together, it tells us that squaring and square rooting are inverse operations, like addition and subtraction, and multiplication and division. So, if we do one and then do the other, the numbers don’t change.

$$\sqrt{9}^2=9$$

$$\sqrt{6^2}=6$$

This is particularly helpful when the numbers don’t have square roots or squares that are easy to find.

$$\sqrt{2.5}^2=2.5$$

$$\sqrt{17}^2=17$$

$$\sqrt{34.56^2}=34.56$$

$$\sqrt{\pi^2}=\pi$$

## Example Questions

$$\sqrt{13.77}^2=13.77$$

$$\sqrt{1521.6^2} =1521.6$$

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