What is 10³ (ten cubed)?

1000. A power means repeated multiplication, and the index 3 counts how many tens to multiply together: 10³ = 10 × 10 × 10 = 1000. The common wrong answer 30 comes from reading it as 10 × 3, multiplying the base by the index once — but that is barely bigger than ten, the tell that the multiplication was never repeated.

What does the index (the small raised number) in a power mean?

It counts how many copies of the base are multiplied together — it is not a number to multiply the base by. In 2⁵ the index 5 means five twos multiplied: 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5 = 10. So to evaluate a power you multiply that many copies of the base; to write a string of equal factors as a power you count how many there are.

Why is 2⁵ different from 2 × 5 and from 5²?

They are three different numbers. 2⁵ is repeated multiplication, 2 × 2 × 2 × 2 × 2 = 32. 2 × 5 is a single product, 10. And 5² swaps the base and index, 5 × 5 = 25. A power repeats the multiplication, a product does it once, and swapping the base and index changes the value, so the three must be kept apart.

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

Powers are repeated multiplication: why 10³ is 1000, not 30

Powers trip students who read $10^3$ as “base × index” — answering $10 \times 3 = 30$ — without noticing what the small raised number is actually telling them to do. The index counts how many copies of the base to multiply together, so $10^3 = 10 \times 10 \times 10 = 1000$ .

The thirty-second fix: a power is repeated multiplication. The index counts how many copies of the base you multiply together — it is not a number to multiply the base by once. So $10^3 = 10 \times 10 \times 10 = 1000$ , never $10 \times 3 = 30$ .

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How to spot it in your own work

You multiplied the base by the index — writing $10^3 = 10 \times 3 = 30$ instead of $10 \times 10 \times 10 = 1000$ .
Your answer came out barely bigger than the base — a sure sign you multiplied once instead of repeating the multiplication.
You turned a string of equal factors into a product — writing $2 \times 2 \times 2 \times 2 = 2 \times 4 = 8$ instead of $2^4 = 16$ .
You confused a power with a product or the swapped form — $2^5$ , $2 \times 5$ , and $5^2$ give 32, 10, and 25, not all the same number.
You read $10^4$ as $10 \times 4 = 40$ instead of $10000$ — one followed by four zeros.

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Jun23 P1 Q3d / Nov24 P1 Q1b shape, non-calculator):

Work out 10³.

The misconception answer is $10 \times 3 = 30$ — multiplying the base by the index. But 30 is barely bigger than ten, which cannot be the value of ten cubed.

Repeat the multiplication: the index 3 counts three tens, so $10^3 = 10 \times 10 \times 10 = 1000$ .

Why students fall for this

Multiplication is the first move students reach for when they see two numbers stacked as a power. But the raised number is an index, and it does not say “multiply by”; it says how many copies of the base to multiply together. Three is not a multiplier in $10^3$ ; it is the count of tens in the product $10 \times 10 \times 10$ . The giveaway is the size: a real power grows fast, so $10^3 = 1000$ , whereas $10 \times 3 = 30$ is barely bigger than the base.

The same belief — that the index is a multiplier — reappears when writing a string of equal factors as a power. Students see $2 \times 2 \times 2 \times 2$ , count four twos, and then write $2 \times 4 = 8$ instead of $2^4 = 16$ . The four is the count of factors, not a number to multiply 2 by.

AQA Foundation papers exploit every face of this on the non-calculator paper — evaluating $10^3$ (Jun23 P1 Q3d), $3^3$ (Nov24 P1 Q1b), and powers inside a larger product such as $2^7 \times 5^2$ (Jun22 P3 Q17a). Each one rewards repeated multiplication and punishes base × index.

The fix: A power is repeated multiplication: the index counts how many copies of the base to multiply together

The index counts the copies of the base. $10^3$ means three tens multiplied: $10 \times 10 \times 10 = 1000$ . If a power comes out barely bigger than the base, you multiplied once instead of repeating the multiplication.

It works the other way too. To write equal factors as a power, count them: $2 \times 2 \times 2 \times 2$ is four twos, so $2^4 = 16$ — never $2 \times 4 = 8$ .

Keep a power apart from a product. $2^5 = 32$ , $2 \times 5 = 10$ , and $5^2 = 25$ are three different numbers — repeat, multiply once, and swap give different sizes.

Worked example

Work out 4³ (four cubed).

Read the index as a count. The 3 says “three fours multiplied together”, not “times 3”.
Repeat the multiplication.
$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$
Sense-check the size. 64 is far bigger than the base; the trap $4 \times 3 = 12$ is barely bigger than 4, so it is wrong.

The same habit handles powers of 10 by the zeros: $10^4 = 10 \times 10 \times 10 \times 10 = 10000$ — one with four zeros, not $10 \times 4 = 40$ . Count the copies, multiply them.

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Common questions

Why can’t I just multiply the base by the index?: Because the index counts how many copies of the base to multiply together, not what to multiply the base by once. $10^3$ is three tens multiplied: $10 \times 10 \times 10 = 1000$ . Multiplying $10 \times 3$ gives 30, barely bigger than ten — impossible for ten cubed.
How do I write 2 × 2 × 2 × 2 as a power?: Count how many equal factors there are. There are four twos, so it is $2^4$ , and $2^4 = 16$ . The four is the count of twos — it is not a number to multiply 2 by, so it is not $2 \times 4 = 8$ .
How do I work out a power of 10 quickly?: The index tells you how many zeros. $10^4$ is one followed by four zeros, $10000$ , because it is $10 \times 10 \times 10 \times 10$ . Writing $10 \times 4 = 40$ is the base × index trap.
Is 2⁵ the same as 5²?: No. $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$ , while $5^2 = 5 \times 5 = 25$ . Swapping the base and the index changes the value, so the two are different numbers.

Related misconceptions

Squaring is not doublingThe n = 2 case of the same trap — 19² is 19 × 19 = 361, not 19 × 2 = 38; squaring multiplies the number by itself.
Index laws: keep the base, add or subtract the indicesBuilds straight on this idea — once a power is repeated multiplication, 2³ × 2⁴ = 2⁷ because you count copies of the base.
Speed, distance, time & rates: convert the time before you divideA neighbouring Foundation skill — another place where the wrong operation feels right until you sense-check the size of the answer.

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