Square numbers and square roots | Year 7

01 / LEARN

Understand the idea.

Squaring a number

To square a number, multiply it by itself. An arrangement with $n$ rows and $n$ dots in each row contains $n^{2}$ dots. This explains the sequence $0, 1, 4, 9, 16, 25, \dots$ of square numbers.

For a square, the area is its side length multiplied by itself. A side measured in centimetres produces an area in square centimetres, ${cm}^{2}$ . Keep the unit of length distinct from the unit of area.

Taking a square root

The principal square root is the non-negative number whose square equals the number inside the root sign. Thus $\sqrt{144} = 12$ . For non-negative inputs, taking a square root reverses squaring; a physical side length is non-negative too.

Use neighbouring perfect squares to locate an unfamiliar root. Since 50 lies between 49 and 64, $\sqrt{50}$ lies between 7 and 8. Rounding it to 7 gives an approximation rather than an exact equality.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Evaluate a square and its root

Calculate $13^{2}$ and $\sqrt{196}$ .

Worked solution

1
Expand the square as a product.
$13^{2} = 13 \times 13 = 169$
The exponent 2 means two factors of 13.
2
Find a number whose square is 196.
$14^{2} = 14 \times 14 = 196$
A square root can be checked by squaring the proposed result.
3
Choose the principal root.
$\sqrt{196} = 14$
The root symbol denotes the non-negative root, so it returns 14.

Answer

$13^{2} = 169$ and $\sqrt{196} = 14$

EXAMPLE 02

Recover a side from an area

A square has area 81 square centimetres. Find its side length.

Worked solution

1
Relate side length to area.
$s^{2} = 81, s \geq 0$
For a square, area equals side length squared; here s is measured in centimetres.
2
Find the matching square number.
$9^{2} = 81$
Nine multiplied by itself gives the required area.
3
State the length with its unit.
$s = \sqrt{81} = 9 cm$
Taking the non-negative root recovers the side length, not an area.

Answer

9 cm

03 / YOUR TURN

Try it yourself.

Check your understanding

Between which two consecutive whole numbers is $\sqrt{70}$ ?

Show answer & reasoning

8 and 9

The neighbouring squares are $8^{2} = 64$ and $9^{2} = 81$ . Since $64 < 70 < 81$ , the non-negative root satisfies $8 < \sqrt{70} < 9$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Calculate $8^{2}$ .
2 marks
Find $\sqrt{121}$ .
2 marks
A square display uses 81 tiles with the same number of tiles in each row and column. How many tiles are in one row?
2 marks
Between which two consecutive whole numbers is $\sqrt{50}$ ?
2 marks
Evaluate $3^{2} + 4^{2}$ .
2 marks
A student says $\sqrt{9 + 16} = 7$ . Explain the error and give the correct value.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Calculate $8^{2}$ .
2 marks
Square means multiply by itself: $8 \times 8 = 64$ .
64
Find $\sqrt{121}$ .
2 marks
$11 \times 11 = 121$ .
Therefore the principal square root is 11.
11
A square display uses 81 tiles with the same number of tiles in each row and column. How many tiles are in one row?
2 marks
Let the number in a row be $n$ .
The square has $n^{2} = 81$ tiles, so $n = 9$ .
9
Between which two consecutive whole numbers is $\sqrt{50}$ ?
2 marks
$7^{2} = 49$ and $8^{2} = 64$ .
Because 50 lies between them, its root lies between 7 and 8.
7 and 8
Evaluate $3^{2} + 4^{2}$ .
2 marks
$3^{2} = 9$ and $4^{2} = 16$ .
$9 + 16 = 25$ .
25
A student says $\sqrt{9 + 16} = 7$ . Explain the error and give the correct value.
3 marks
Add inside the root first: $9 + 16 = 25$ .
$\sqrt{25} = 5$ .
Square roots do not generally distribute over addition.
The correct value is 5.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Place value and powers of ten; efficient whole-number calculations and order of operations; factors, multiples and primes; prime factorisation and exponent notation; square numbers and square roots; estimation and rounding.

Official curriculum source ↗

In this lesson

Understand the idea.

Squaring a number

Taking a square root

See the method in action.

Evaluate a square and its root

Expand the square as a product.

Find a number whose square is 196.

Choose the principal root.

Recover a side from an area

Relate side length to area.

Find the matching square number.

State the length with its unit.

Try it yourself.

Check your understanding

Keep practising