Year 7AC9M7N01

Square Numbers and Square Roots

Describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems.

Squaring and square rooting undo each other. Times and divide work the same way.

Builds onYr 6 · Multiplication facts

Why this works

The words come straight from geometry, and holding the picture in mind makes everything else obvious.

A square with sides of 4 contains

4 \times 4 = 16

small squares. That is why we call 16 a square number and write

4^{2}

4^{2} = 16

Going backwards: if a square has an area of 16, its side must be 4. That is the square root.

\sqrt{16} = 4

So the two operations are inverses — each undoes the other.

\sqrt{4^{2}} = 4

A number is a perfect square only if it makes a whole-sided square. $16$ does; $17$ does not, so $\sqrt{17}$ is not a whole number. Knowing the first fifteen square numbers by heart makes the rest of school algebra noticeably easier.

Worked example

e.g.

\sqrt{144} + 3^{2}

Deal with the root. Ask: what number times itself gives 144? $12 \times 12 = 144 \ \text{ so } \ \sqrt{144} = 12$
Deal with the power. $3^{2} = 3 \times 3 = 9$
Add the results. $12 + 9 = 21$

Common mistakes

Wrong

5^{2} = 10

Right

5^{2} = 5 \times 5 = 25

Doubling instead of squaring. The small 2 means 'use it twice as a factor', not 'multiply by 2'.

Wrong

\sqrt{36} = 18

Right

\sqrt{36} = 6

Halving instead of rooting. Ask what number times itself gives 36, not what is half of it.

Wrong

\sqrt{9 + 16} = 3 + 4 = 7

Right

\sqrt{9 + 16} = \sqrt{25} = 5

Roots do not split across addition. Work out what is under the root sign first, then take the root of the total.

Practice

FluencyGet quick and accurate at the method.

7^{2}

Answer

49

12^{2}

Answer

144

\sqrt{64}

Answer

8

\sqrt{121}

Answer

11

9^{2} - \sqrt{25}

Answer

76

\sqrt{100} + 4^{2}

Answer

26

\sqrt{81} \times 2^{2}

Answer

36

15^{2}

Answer

225

\sqrt{169}

Answer

13

\sqrt{49 + 32}

Answer

9

AppliedThe same maths, inside a real question.

A square courtyard has an area of 196 m². How long is each side?

Answer14 m, since

\sqrt{196} = 14

A square photo has 9 cm sides. What is its area?

Answer81 cm², since

9^{2} = 81

A square garden has an area of 225 m². How much fencing is needed to go all the way around it?

Answer60 m. The side is

\sqrt{225} = 15

m, and the perimeter is

4 \times 15 = 60

Tiles are 1 m squares. Mia has 150 of them. What is the largest square she can tile, and how many tiles are left over?

AnswerA

12 \times 12

square, using 144 tiles and leaving 6 over.

12^{2} = 144

and

13^{2} = 169

, which is too many.