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Exponents and roots

An exponent is shorthand for repeated multiplication, and reading it as shorthand is most of the battle. Children who see 4³ and think "four times three" are not confused about exponents; they are reading the notation as though the little number were an ordinary one, which is a reasonable mistake to make once and worth correcting immediately.

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Reading an exponent

4 cubed

  1. The little 3 counts how many 4s are multiplied, not what to multiply by.
  2. So it is 4 x 4 x 4.
  3. 4 x 4 is 16, and 16 x 4 is 64.
  4. 4 cubed is 64.

Worth knowing. 4 x 3 is 12 and 4 cubed is 64. It is worth writing both down side by side once, because a child who has seen the difference stops making it.

Finding a square root

What is the square root of 49?

  1. The question asks: what number times itself makes 49?
  2. Work up through the squares: 5 x 5 is 25, 6 x 6 is 36, 7 x 7 is 49.
  3. So 7 is the number.
  4. The square root of 49 is 7.

Worth knowing. Knowing the squares to 12 by heart turns this from a search into a lookup, which is why the squares are worth learning as a set rather than met one at a time.

Where it usually goes wrong

  • Multiplying the base by the exponent, so 4 cubed comes out as 12.
  • Losing count of the multiplications in a higher power — 2 to the fifth is five 2s, not four.
  • Giving a root that is close but not exact, because the squares were not checked.

The Professor says it in full the first few times: four cubed is four times four times four. The notation gets shorter later; the meaning does not.

Practise it

All level paths

Questions teachers and parents ask

When are exponents taught?

Writing and evaluating numerical expressions with whole-number exponents is grade 6 (6.EE.A.1), and square and cube root symbols arrive in grade 8 (8.EE.A.2). Powers of ten turn up earlier, in grade 5, because they explain what happens to the zeros when you multiply.

Why do the squares come first?

Because they are the ones that turn up everywhere afterwards — in areas, in roots, in the Pythagorean theorem, in quadratics. Knowing the squares to 12 by heart is worth as much as knowing the times tables, and for the same reason.

Are square roots always whole numbers?

No, and most are not — the square root of 50 is about 7.07 and never stops. Every root asked for here is a perfect square with a whole answer, because this is the topic where the idea is introduced rather than the one where decimals are.

Where this shows up

Standards

  • 6.EE · Expressions and equations
  • 8.NS · The number system
  • 8.EE · Expressions and equations
  • CCSS.MATH.CONTENT.5.NBT.A.2 — Explain the pattern in the zeros of a product when multiplying by a power of 10, and use whole-number exponents to write powers of 10.
  • CCSS.MATH.CONTENT.6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
  • CCSS.MATH.CONTENT.8.EE.A.2 — Use square root and cube root symbols to represent and solve equations of the form x squared equals p and x cubed equals p.

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