Indices (Exponents) for a 15-year-old — clear rules, examples, negatives & fractions

Learn indices (exponents) step-by-step for a 15-year-old: notation, laws (product, quotient, power), zero and negative indices, fractional indices, worked examples, common mistakes and practice problems with answers.

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What are indices (exponents)?

An index (also called an exponent or power) tells you how many times to multiply a number by itself. If you see 23, it means 2 × 2 × 2 = 8. In notation, the base is the main number (2) and the index/exponent is the small number written above and to the right (3).

Basic notation

  • an means multiply a by itself n times: a × a × ... × a (n factors).
  • Examples: 52 = 25, 34 = 81, 71 = 7.

Important rules (laws) of indices — use these to simplify expressions

  • Product rule: am × an = am+n (same base — add exponents).
  • Quotient rule: am ÷ an = am−n (same base — subtract exponents).
  • Power rule: (am)n = am×n (multiply exponents).
  • Power of a product: (ab)n = anbn.
  • Power of a quotient: (a/b)n = an/bn (b ≠ 0).
  • Zero index: a0 = 1 (for a ≠ 0).
  • Negative index: a−n = 1 / an (a ≠ 0).
  • Fractional index: a1/n = n'th root of a (e.g., a1/2 = √a). More generally, am/n = (a1/n)m = (am)1/n.

Worked examples — step by step

1) Simplify 23 × 24:

Same base (2), so add exponents: 23+4 = 27 = 128.

2) Simplify 54 ÷ 52:

Subtract exponents: 54−2 = 52 = 25.

3) Simplify (32)3:

Multiply exponents: 32×3 = 36 = 729.

4) Evaluate 4−2:

Negative index: 4−2 = 1 / 42 = 1/16.

5) Evaluate 161/2:

Fractional index: 161/2 = √16 = 4. Or 271/3 = 3 (cube root).

6) Simplify (2×3)2:

Use power of a product: (2×3)2 = 22 × 32 = 4 × 9 = 36.

Common mistakes to watch for

  • Don’t add exponents when bases are different (23 × 32 stays as is unless you calculate numbers).
  • ab+c is not equal to ab + ac (exponents don’t distribute over addition).
  • Remember a0 = 1 only if a ≠ 0.
  • With negative exponents, rewrite as reciprocals before simplifying fractions.

Quick reference (examples)

  • 32 × 35 = 37
  • 100 = 1
  • 2−3 = 1/8
  • 91/2 = 3
  • (x2y)3 = x6y3

Practice problems (try before checking answers)

  1. Simplify 73 × 7−1.
  2. Simplify (42)3.
  3. Find 813/4.
  4. Simplify (2/5)−2.

Answers

  1. 73−1 = 72 = 49.
  2. 42×3 = 46 = 4096.
  3. 813/4 = (811/4)3. 81 = 34, so 811/4 = 3, then 33 = 27.
  4. (2/5)−2 = (5/2)2 = 25/4.

Final tips

  • When possible, rewrite expressions to use the index rules — that makes simplification easier.
  • Practice with numbers first, then apply to algebraic expressions (with letters like x, y).
  • Use a calculator for big powers, but understand the rules so you can simplify before calculating.

If you want, I can give more practice problems or explain how indices show up in scientific notation and graphs — tell me which you prefer.

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