Practice Maths

Indices and Index Laws

Key Terms

An index (or exponent) tells how many times the base is multiplied by itself: an = a × a × … × a (n times).
Index laws allow us to simplify expressions without expanding fully.
The zero index law: any non-zero base raised to the power 0 equals 1.
The negative index law: a−n = 1/an — a negative index means “take the reciprocal”.
The fractional index law: a1/n = n√a — a fractional index means “take the root”.
Scientific notation
: write a number as m × 10n where 1 ≤ m < 10 and n is an integer.
Index Laws Summary
Multiplicationam × an = am+nAdd exponents (same base)
Divisionam ÷ an = am−nSubtract exponents (same base)
Power of a power(am)n = amnMultiply exponents
Zero indexa0 = 1  (a ≠ 0)Any non-zero base to the power 0
Negative indexa−n = 1/anReciprocal
Fractional indexa1/n = n√a   and   am/n = (n√a)mnth root

Powers of 2 — each step multiplies by 2

Exponent 2n Value 0 20 1 1 21 2 2 22 4 3 23 8 4 24 16 ×2 ×2 ×2 ×2
Hot Tip Index laws only apply when the bases are the same. For example, 23 × 32 cannot be simplified using index laws — they have different bases. Also remember: a negative index does not make the result negative! 2−3 = 1/8, not −8.

Worked Example 1 — Simplifying with index laws

Question: Simplify   (a) 34 × 32   (b) (x3)4 ÷ x5   (c) 5x−2 × 2x3

(a) 34 × 32 = 34+2 = 36 = 729

(b) (x3)4 ÷ x5 = x12 ÷ x5 = x12−5 = x7

(c) 5 × 2 × x−2+3 = 10x

Worked Example 2 — Fractional indices and scientific notation

Question: Evaluate   (a) 272/3   (b) Write 0.00045 in scientific notation   (c) Write 3.6 × 104 as an ordinary number.

(a) 272/3 = (3√27)2 = 32 = 9

(b) 0.00045 = 4.5 × 10−4

(c) 3.6 × 104 = 3.6 × 10000 = 36 000

Why the Index Laws Are True — Deriving Each One

Every index law follows directly from the definition of a positive integer exponent: an means a multiplied by itself n times. The multiplication law am × an = am+n works because you are combining m copies and n copies of a, giving m+n copies in total. For example, 23 × 24 = (2×2×2) × (2×2×2×2) = 27. The division law am ÷ an = am−n follows because you are cancelling n copies from the m copies, leaving m−n. The power of a power law (am)n = amn works because each of the n groups contains m copies of a, giving mn copies. These three laws form the foundation; the others are derived from them.

The Zero Index and Negative Index — Why a&sup0; = 1 and a−n = 1/an

Applying the division law to am ÷ am gives am−m = a0. But any non-zero quantity divided by itself equals 1. Therefore a0 = 1 for all a ≠ 0. Notice that 00 is undefined (a separate mathematical question). For the negative index: applying the division law, a0 ÷ an = a−n. But a0/an = 1/an. Therefore a−n = 1/an. A critical misconception to avoid: a−2 is NOT −a2. The negative in the index means “reciprocal”, not “negative value.” For instance, 3−2 = 1/9, which is positive.

Exam Tip: Index laws only apply when the bases are identical. You cannot simplify 23 × 54 using index laws because 2 ≠ 5. Similarly, x2 + x3 cannot be simplified using index laws — addition of powers with the same base does not simplify. Only multiplication (am × an) and division (am ÷ an) of powers with the same base allow you to combine indices.

Fractional Indices and Their Connection to Roots

The fractional index law a1/n = n√a arises from demanding consistency with the power of a power law. If a1/2 is defined, then (a1/2)2 = a1 = a. But √a squared also equals a. So a1/2 must equal √a. More generally, am/n = (a1/n)m = (n√a)m. It is often easier to take the root first (before the power) to keep numbers smaller. For example, 272/3: the cube root of 27 is 3, and 3² = 9. Trying to compute 27² = 729 first and then take the cube root is harder. Always root first, power second.

Solving Exponential Equations by Matching Bases

When both sides of an equation can be written as powers of the same base, you equate the exponents. For 4x = 8: rewrite as (22)x = 23, giving 22x = 23, so 2x = 3 and x = 3/2. The key skill is recognising common bases. The most useful are powers of 2 (4, 8, 16, 32, 64), powers of 3 (9, 27, 81), and powers of 5 (25, 125). When the equation involves a fraction, remember that a−n = 1/an: so 3x = 1/9 = 3−2 gives x = −2. If bases cannot be matched, logarithms are required (Year 11/12 topic).

Exam Tip: When simplifying expressions with multiple index laws, work systematically: apply power of a power first, then multiplication/division, then handle negative indices at the very end by moving factors between numerator and denominator. Final answers should always have positive indices unless the question specifies otherwise.

Scientific Notation and Why It Uses Powers of 10

Scientific notation a × 10n (where 1 ≤ a < 10) is powerful because all the index laws apply directly. Multiplying (3 × 104) × (2 × 103) = (3×2) × 104+3 = 6 × 107. Dividing (8 × 107) ÷ (4 × 103) = 2 × 104. If the coefficient falls outside [1, 10), rewrite: 12 × 105 = 1.2 × 106. The condition 1 ≤ a < 10 ensures the representation is unique, which is why scientific notation is universally used in science and engineering to avoid ambiguity.

Mastery Practice

See Answers ➔
  1. Fluency

    Evaluate each expression without a calculator.

    1. (a) 25
    2. (b) 34
    3. (c) 50
    4. (d) 4−2
    5. (e) 81/3
    6. (f) 163/4
  2. Fluency

    Apply the multiplication law to simplify each expression.

    1. (a) 53 × 54
    2. (b) x7 × x2
    3. (c) 3a4 × 4a3
    4. (d) 2m5 × 3m−2
  3. Fluency

    Apply the division law to simplify each expression.

    1. (a) 68 ÷ 63
    2. (b) x9 ÷ x4
    3. (c) 12a6 ÷ 4a2
    4. (d) 15x3 ÷ 5x5
  4. Fluency

    Apply the power of a power law to simplify.

    1. (a) (23)4
    2. (b) (x5)2
    3. (c) (3a2)3
    4. (d) (2x3y2)4
  5. Fluency

    Write each number in scientific notation.

    1. (a) 3 800 000
    2. (b) 0.000 072
    3. (c) 450 000 000
    4. (d) 0.0000035
  6. Understanding

    Simplify each expression, writing your answer with positive indices only.

    Method: Apply index laws step by step. At the end, rewrite any negative indices as positive by moving factors to the other part of the fraction.
    1. (a) (x2y3)2 × x−3y
    2. (b) (4a3b−2) ÷ (2a−1b3)
    3. (c) (3x2)3 ÷ (9x4)
    4. (d) (m−1n2)3 × (m4n−3)2
  7. Understanding

    Express each of the following using a single index.

    1. (a) 4√(x3)
    2. (b) 1 / (3√x)
    3. (c) (√x)5
    4. (d) 3√(8x6)
  8. Understanding

    Solve for x in each equation by matching bases.

    Method: Write both sides as powers of the same base, then equate the exponents. e.g. 2x = 8 → 2x = 23 → x = 3.
    1. (a) 3x = 81
    2. (b) 2x+1 = 16
    3. (c) 52x−1 = 25
    4. (d) 4x = 8 (write 4 and 8 as powers of 2)
  9. Understanding

    Operations with scientific notation.

    1. (a) (3 × 104) × (2 × 103)
    2. (b) (8 × 107) ÷ (4 × 103)
    3. (c) (5 × 10−3) × (6 × 105)
    4. (d) The distance from Earth to the Sun is approximately 1.5 × 1011 m. Light travels at 3 × 108 m/s. How many seconds does light take to travel from the Sun to Earth?
  10. Problem Solving

    Simplifying complex expressions.

    Challenge. Apply multiple index laws carefully, showing all steps.
    1. (a) Simplify: [(2x3y−1)2 × (4x−1y3)] ÷ [8x2y4]
    2. (b) Simplify: (27a6b3)2/3 ÷ (3a2b)
    3. (c) Show that (am+n × am−n) / a2m = 1 for all values of a (a ≠ 0) and all values of m and n.
  11. Problem Solving

    Indices in context.

    Real-world application.
    1. (a) A bacteria culture starts with 500 bacteria and doubles every hour. Write an expression for the number of bacteria after n hours. How many bacteria are there after 8 hours?
    2. (b) The mass of a proton is approximately 1.67 × 10−27 kg and the mass of an electron is approximately 9.11 × 10−31 kg. How many times heavier is a proton than an electron? Give your answer to 3 significant figures in scientific notation.
    3. (c) If 4n = 1024, find n. Then find the value of 4n × 8n/3.