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Mathematics 05801.3

Indices and standard form

The laws of indices and writing very large or small numbers.

Learning objectives

What you need to be able to do

Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.

  • 1.3.1Use the laws of indices, including negative and fractional indices.
  • 1.3.2Convert between ordinary numbers and standard form and calculate with them.

6 minute read

Indices and standard form

The laws of indices

All of these follow from what an index means — how many times the base is multiplied by itself.

  • aᵐ × aⁿ = aᵐ⁺ⁿ — add the powers when multiplying.
  • aᵐ ÷ aⁿ = aᵐ⁻ⁿ — subtract when dividing.
  • (aᵐ)ⁿ = aᵐⁿ — multiply when raising a power to a power.
  • a⁰ = 1 for any non-zero a.
  • a⁻ⁿ = 1/aⁿ — a negative index means reciprocal, not a negative answer.
  • a^(1/n) = ⁿ√a, so a^(m/n) = (ⁿ√a)ᵐ.

The two that cause most errors: a negative index never makes the answer negative, and a⁰ = 1 rather than 0.

Standard form

A number in standard form is written a × 10ⁿ where 1 ≤ a < 10 and n is an integer.

  • Large numbers give a positive n: 4 500 000 = 4.5 × 10⁶.
  • Small numbers give a negative n: 0.00032 = 3.2 × 10⁻⁴.

The condition 1 ≤ a < 10 is strict: 45 × 10⁵ is not in standard form even though it is numerically correct, and it loses the mark.

Calculating in standard form

Multiply or divide the number parts, then apply the index laws to the powers of ten. Afterwards, check a is still between 1 and 10 and adjust if not — this final adjustment is the step most often forgotten.

Think of it like this

Standard form is scientific shorthand for scale: the power of ten tells you how many places the decimal point has moved, so a positive power means "this is big" and a negative one means "this is small".

Worked examples

Method, step by step

Calculate (3 × 10⁵) × (4 × 10⁻²), giving your answer in standard form.

  1. 1Multiply the number parts: 3 × 4 = 12.
  2. 2Apply the index law to the powers of ten: 10⁵ × 10⁻² = 10⁵⁺⁽⁻²⁾ = 10³.
  3. 3This gives 12 × 10³, but 12 is not between 1 and 10.
  4. 4Rewrite 12 as 1.2 × 10¹, so the answer becomes 1.2 × 10⁴.

1.2 × 10⁴

Evaluate 16^(3/4).

  1. 1A fractional index m/n means take the nth root, then raise to the power m.
  2. 2The denominator 4 means the fourth root: ⁴√16 = 2.
  3. 3The numerator 3 means cube it: 2³ = 8.

8

Common misconceptions

  • Thinking `2⁻³` is negative. It equals 1/8 — a positive number. The minus sign inverts, it does not negate.
  • Writing an answer like `45 × 10⁵` and calling it standard form. The first part must satisfy 1 ≤ a < 10, so it should be `4.5 × 10⁶`.
  • Assuming `a⁰ = 0`. Any non-zero number to the power zero is 1.

In the exam

  • After any standard form calculation, look at the number part and fix it if it is not between 1 and 10 — that adjustment is a mark on its own.
  • For fractional indices, deal with the root first and the power second; the numbers stay smaller and mistakes are less likely.