Square Numbers

Know the square numbers, spot them in a list, square a decimal, and find what to multiply by to make a number square.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 26 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What a Square Number Is

A square number is a whole number times itself. 5 rows of 5 dots make a square with 25 dots, so 25 is a square number.

Formula

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

The small 2 means "squared": use the number twice in a multiplication.

Memory trick

Learn the first fifteen: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.

Key idea

The gaps between square numbers are the odd numbers: 1 to 4 is +3, 4 to 9 is +5, 9 to 16 is +7. So the square numbers grow faster and faster.

Common mistake

Reading $5^{2}$ as 5 × 2 = 10. Squared means 5 × 5 = 25.

Spot a Square Number

The 1-mark question you see most: “From the list, write down a square number”, or “Write down a square number between 50 and 70”.

From 12, 18, 36, 45, 50, 63, write down a square number.

Worked example

1
$1, 4, 9, 16, 25, 36, 49, 64$ — Write the squares you know in order, up to just past 63, the largest number in the list.
2
$36 = 6 \times 6$ — 36 is in both lists, and 6 × 6 = 36. So 36 is a square number.
3
$12, 18, 45, 50, 63$ — Each of these sits between two squares (for example 45 is between 36 and 49), so none of them is square.

Some questions give a range instead: a square number between 50 and 70.

Worked example

1
$7^{2} = 49$ — Too small: 49 is just below 50, so try the next whole number.
2
$8^{2} = 64$ — 64 is between 50 and 70, so it is an answer.
3
$9^{2} = 81$ — Check the next one: 81 is above 70. So 64 is the only square in the range.

Common mistake

Picking 50 because 5 × 10 = 50. A square number is one number times itself, so check the squares you know.

Common mistake

Judging a √ by the number under it. Work it out first: $144 = 12$ , which is not square, but $81 = 9$ , which is.

Key idea

If you must write down all the square numbers up to 16, start at 1 and include 16: 1, 4, 9, 16. Missing 1 is the usual slip.

Square a Decimal

The 1-mark question: “Find the value of $2. 6^{2}$ ”. Multiply 2.6 by itself, not by 2.

Worked example

1
$2. 6^{2} = 2.6 \times 2.6$ — Squared means the number times itself, so write 2.6 twice.
2
$26 \times 26 = 676$ — Ignore the decimal points for now and multiply the digits: 26 × 20 = 520, 26 × 6 = 156, and 520 + 156 = 676.
3
$2.6 \times 2.6 = 6. 76$ — Each 2.6 has one decimal place, so the answer has 1 + 1 = 2. Count two places in from the right of 676.

Common mistake

Working out 2.6 × 2 = 5.2. Squared never means times 2.

Common mistake

Writing $0. 3^{2} = 0.9$ . 3 × 3 = 9, and two decimal places make it 0.09.

Key idea

Check the size: 2.6 is between 2 and 3, so its square is between 4 and 9. 6.76 fits.

Negatives and Fractions

Questions also square a negative number or a fraction. Each needs one extra move.

A negative number

Worked example

1
$(- 6)^{2} = (- 6) \times (- 6)$ — Keep −6 in brackets, so the minus is squared too.
2
$(- 6) \times (- 6) = 36$ — Minus times minus is plus. So a square is never negative.

A fraction

Worked example

1
$(\frac{3}{4})^{2} = \frac{3 ^{2}}{4 ^{2}}$ — The square acts on the whole fraction, so square the top and the bottom.
2
$= \frac{9}{16}$ — 3 × 3 = 9 and 4 × 4 = 16.

Common mistake

Typing −6² on a calculator gives −36, because it squares only the 6. Type the brackets: $(- 6)^{2} = 36$ .

Common mistake

Writing $(\frac{3}{4})^{2} = \frac{9}{4}$ . Square the bottom as well: 16, not 4.

Make It a Square

The 1-mark question: “Find the smallest whole number that 60 can be multiplied by to give a square number.” Prime factors make it quick, because in a square every prime comes in pairs.

Worked example

1
$60 = 2 \times 2 \times 3 \times 5$ — Split 60 into primes first: 60 = 2 × 30, 30 = 2 × 15, 15 = 3 × 5. The next card shows the tree.
2
$60 = 2^{2} \times 3 \times 5$ — Group equal primes. A square is n × n, so each prime in it appears an even number of times.
3
$2^{2} = 2 \times 2$ — The two 2s are a pair, so they need nothing.
4
$3, 5$ — The 3 and the 5 are on their own, so each needs one more as a partner.
5
$3 \times 5 = 15$ — Multiply only the missing partners. The smallest number is 15.
6
$60 \times 15 = 900 = 2^{2} \times 3^{2} \times 5^{2} = 3 0^{2}$ — Check: every prime is now in a pair, and 2 × 3 × 5 = 30, so 900 = 30 × 30.

Key idea

Any even power is already pairs: $2^{4} = 2^{2} \times 2^{2}$ . For $2^{4} \times 3 \times 7^{2}$ , only the 3 is alone, so multiply by 3.

Common mistake

Answering 60, because 60 × 60 is square. It works, but it is not the smallest. Only the primes with no partner are needed.

Key idea

This part usually follows “write the number as a product of primes”, which has marks for working. Write the factor tree or division steps down, then use that line here.

Square Factors

Two more questions use the same pairs: write a number as two square numbers multiplied, or find its largest square factor.

Write 1225 as a product of two square numbers, both greater than 1

Worked example

1
$1225 = 5^{2} \times 7^{2}$ — Primes first: 1225 = 5 × 245, 245 = 5 × 49, and 49 = 7 × 7.
2
$5^{2} = 25, 7^{2} = 49$ — Each pair is a square number on its own: 25 and 49.
3
$1225 = 25 \times 49$ — So 1225 is 25 times 49. Both are square and both are bigger than 1.

Find the largest square number that is a factor of 72

Worked example

1
$72 = 2^{3} \times 3^{2}$ — Primes first: 72 = 8 × 9 = 2 × 2 × 2 × 3 × 3.
2
$2^{3} = 2^{2} \times 2$ — Take out as many pairs as you can. Three 2s give one pair and one spare 2.
3
$2^{2} \times 3^{2} = 4 \times 9 = 36$ — Multiply all the pairs together: the 2s and the 3s. That is the largest square factor.
4
$72 = 36 \times 2$ — Check: 72 ÷ 36 = 2, a whole number, and 36 = 6 × 6.

Common mistake

Giving 4 or 9. Both are square factors of 72, but not the largest. Put all the pairs together.

Show It Is Square and Cube

The 2-mark question: “Show that 729 is both a square number and a cube number.” Write one multiplication for each.

Worked example

1
$27 \times 27 = 729$ — For a square, find a number times itself that makes 729: 27 × 20 = 540, 27 × 7 = 189, and 540 + 189 = 729.
2
$729 = 27^{2}$ — One number times itself, so 729 is a square number: 27 squared.
3
$9 \times 9 = 81$ — For a cube, use one number three times. Multiply the first two nines.
4
$81 \times 9 = 729$ — Multiply by the third nine. It makes 729, so 729 is 9 cubed.
5
$729 = 2 7^{2} = 9^{3}$ — Write both facts together as your answer. Both are shown, so it is both.

Common mistake

Only saying “it is in the list of squares”. Show questions need the working: write the multiplications out.

Key idea

The marks are for working: one correct multiplication earns the first mark, and showing both earns the second.

Which Method? Read the Question

Look for the key word, then use the right lesson.

You see	You do
Write down a square number	check the squares you know
Find the value of 2.6 squared	multiply it by itself
A negative or a fraction squared	brackets; square the top and the bottom
Smallest number to give a square	prime factors, pair them up
Two square numbers, or largest square factor	prime factors, put the pairs together
Square number and cube number	one multiplication for each
Square root, or the √ sign	Square Roots
Cube number, used three times	Cube Numbers
Cube root	Cube Roots
Product of prime factors	Prime Factorization

Key idea

Squares grow fast: 10 squared is 100, 15 squared is 225. If your answer is far off that, check it.

Practise square numbers now

Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.

Practise free on Lemmara

Real past-paper questions on square numbers

Each with a full worked solution.

All 26 questions on this topic →

Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.