What is 5² — is it 10?

No. 5² is 25, not 10. The small raised 2 means use the 5 twice, multiplied together: 5 × 5 = 25. The answer 10 comes from doubling, 5 × 2, which reads the raised 2 as 'times 2' — but it means how many of the number to multiply together, not what to multiply it by.

Why is 19² equal to 361 and not 38?

Because squaring means multiplying the number by itself: 19² = 19 × 19 = 361. The answer 38 is 19 × 2, doubling. Doubling and squaring are different operations and give the same answer at only one number — 2, where 2 × 2 = 2 + 2 = 4. Everywhere else, squaring grows much faster.

When does squaring give the same answer as doubling?

Only at 2. There, 2² = 2 × 2 = 4 and 2 doubled = 2 + 2 = 4, so the two coincide. It is a one-off coincidence, not a rule — at every other number squaring (times itself) and doubling (times 2) differ, for example 6² = 36 but 6 doubled is 12.

GCSE Maths Foundation · AQA · Indices, powers & standard form

Squaring is not doubling: a² means a × a, not a × 2

Squaring trips students who read the small raised 2 as “times 2”. Asked for 5², they answer 10 — $5 \times 2$ — instead of $5 \times 5 = 25$ . The raised 2 counts how many times the number is multiplied by itself, so 19² is $19 \times 19 = 361$ , not $19 \times 2 = 38$ .

The thirty-second fix: to square a number, multiply it by itself. The raised 2 means use the number twice, so a² = a × a, never a × 2. Squaring and doubling agree at only one number — 2, where 2 × 2 = 2 + 2 = 4.

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How to spot it in your own work

You worked a square by doubling — writing $5^2 = 5 \times 2 = 10$ instead of $5 \times 5 = 25$ .
Your answer to a² was about twice the number, not the number times itself — a sure sign you doubled.
You squared a decimal and got a whole number — e.g. $1.5^2 = 3$ instead of the correct $1.5 \times 1.5 = 2.25$ .
In a substitution you doubled the value — $a^2 = 2 \times 25 = 50$ instead of $25 \times 25 = 625$ .
You assumed squaring always matches doubling because it works for 2 — but $6^2 = 36$ , not 12.

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Nov24 P3 Q1b shape, non-calculator):

Work out 19².

The misconception answer is $19 \times 2 = 38$ — doubling the number. But the raised 2 means use the 19 twice, multiplied together.

Square it properly: $19^2 = 19 \times 19 = 361$ .

Why students fall for this

The small raised 2 sits where a multiplier often does, so students read “5²” as “5, times 2” and double it. But the index counts copies of the number: 5² means two 5s multiplied, $5 \times 5 = 25$ . Doubling adds the number to itself, $5 + 5 = 10$ — a different operation entirely.

The trap survives because it gives the right answer at one number. At 2, $2 \times 2 = 4$ and $2 + 2 = 4$ , so squaring and doubling coincide. A student who only ever checks against 2 is never contradicted — yet at 6 the answers are 36 and 12, and at 9 they are 81 and 18.

AQA Foundation papers exploit every face of this — squaring an integer (Nov24 P3 Q1b: 19²), squaring a decimal (Nov24 P1 Q19: 1.5²), and squaring inside a substitution (Jun22 P3 Q12b: a² where a = 25 gives 625, with 50 the doubling trap).

The fix: To square a number, multiply it by itself — the raised 2 means use it twice, never times 2

The raised 2 means “use the number twice, multiplied”. So $5^2 = 5 \times 5 = 25$ , not $5 \times 2 = 10$ . If your answer is about twice the number, you doubled instead of squared.

It works the same for decimals and substituted values. $1.5^2 = 1.5 \times 1.5 = 2.25$ (not 3), and a² at a = 25 is $25 \times 25 = 625$ (not 50).

Squaring and doubling agree only at 2. There $2 \times 2 = 2 + 2 = 4$ . Everywhere else they differ — $6^2 = 36$ while 6 doubled is 12 — so do not let the 2-case fool you.

Worked example

Work out 19².

Read the raised 2 as “times itself”. 19² means two 19s multiplied together: $19 \times 19$ .
Multiply the number by itself.
$19^2 = 19 \times 19 = 361$
Sense-check against doubling. Doubling would give $19 \times 2 = 38$ — far too small. 361 is much bigger, which is what squaring does.

The same rule handles every shape of number: $1.5^2 = 1.5 \times 1.5 = 2.25$ and, for a = 25, $a^2 = 25 \times 25 = 625$ — never the doubling answers 3 and 50.

Find out if this is costing you marks

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Common questions

What does the small raised 2 actually mean?: It counts how many copies of the number to multiply together. So 5² means two 5s, $5 \times 5 = 25$ . It does not mean “times 2”; multiplying by 2 is doubling, a different operation that gives 10.
How do I square a decimal like 1.5?: The same way: multiply it by itself. $1.5^2 = 1.5 \times 1.5 = 2.25$ . A squared decimal need not be a whole number, so 2.25 is fine. Doubling, $1.5 \times 2 = 3$ , is the trap.
If a = 25, what is a²?: $a^2 = a \times a = 25 \times 25 = 625$ . Substituting a value does not change what the raised 2 asks for — multiply it by itself. The common slip is to double the value, $2 \times 25 = 50$ .
Isn’t squaring the same as doubling?: Only for the number 2, where $2 \times 2 = 2 + 2 = 4$ . That is a coincidence, not a rule. At every other number they differ: $6^2 = 36$ but 6 doubled is 12, and $9^2 = 81$ but 9 doubled is 18.

Related misconceptions

Powers are repeated multiplication, not base × indexThe higher-power sibling — 10³ is 10 × 10 × 10 = 1000, not 10 × 3 = 30; the index counts copies, it is not a multiplier.
Index laws: keep the base, add or subtract the indicesThe index-laws companion — 2³ × 2⁴ = 2⁷; operate on the indices, never on the base.

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