Square numbers and square roots | Year 7

01 / LEARN

The idea, step by step

Squaring a number

The square of a number is the product of the number with itself. The square numbers begin 0, 1, 4, 9, 16, 25, 36 and 49. A square arrangement of dots can show why $n^{2}$ counts $n$ rows of $n$ objects.

A square with side length 12 cm has area $12^{2} = 144$ square centimetres. Keep the distinction between a length in cm and an area in ${cm}^{2}$ .

Taking a square root

The principal square root asks which non-negative number squares to give the amount inside the root sign. Thus $\sqrt{144} = 12$ . The square root reverses squaring for non-negative numbers.

When the number is not a perfect square, neighbouring square numbers locate its root. Since $7^{2} = 49$ and $8^{2} = 64$ , the square root of 50 lies between 7 and 8. It is not exactly 7.

A common trap

$\sqrt{9 + 16} = \sqrt{9} + \sqrt{16}$ .

Instead: First add inside the root: $\sqrt{25} = 5$ . Adding the separate roots would give 7, which is not equal.

02 / WORKED EXAMPLES

See the thinking

Example 1

Evaluate a square and its root

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

$13^{2} = 13 \times 13 = 169$ .
Because $14 \times 14 = 196$ , its principal square root is 14.

13^{2} = 169

and

\sqrt{196} = 14

Example 2

Recover a side from an area

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

For a square, area equals side length squared.
Find the non-negative number that squares to 81.
$9^{2} = 81$ .

9 cm

Check your understanding

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

Show answer

8 and 9

Since $8^{2} = 64 < 70 < 81 = 9^{2}$ , its root lies between 8 and 9.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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 ↗

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