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Fundamentals of Business Mathematics and Statistics · Indices and Logarithms

Laws of Indices: Rules, Formulas and Examples

Updated 10 October 2026 · Fact-checked

The laws of indices are rules for simplifying powers. In aᵐ, a is the base and m is the index. For the same base, add indices when multiplying and subtract when dividing. Multiply indices when a power is raised to a power. Also a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root of a.

Understand Laws of Indices

A power is repeated multiplication. In 2⁴, the number 2 is the base and 4 is the index (also called exponent or power). So 2⁴ = 2 × 2 × 2 × 2 = 16.

The laws come straight from this meaning. Take 2³ × 2² = (2×2×2) × (2×2) = 2⁵. You are just counting how many times 2 is multiplied. That is why indices are added when you multiply powers of the same base.

Division works the same way. 2⁵ ÷ 2² cancels two 2s from the top and bottom, leaving 2³. So you subtract the indices. If you push this rule further, 2³ ÷ 2³ = 2⁰, but any number divided by itself is 1. So a⁰ = 1 for a ≠ 0.

Negative indices follow the same pattern. 2² ÷ 2³ = 2⁻¹, and directly it equals 1/2. So a⁻ⁿ = 1/aⁿ. A negative index means reciprocal, not a negative number.

A fractional index means a root. Since (a^(1/2))² = a, a^(1/2) is the square root of a. In general a^(1/n) is the nth root of a, and a^(m/n) = (a^(1/n))ᵐ. Take the root first, then the power, to keep numbers small.

Key formulas to remember

Product law
aᵐ × aⁿ = aᵐ⁺ⁿ
Same base only. Add the indices.
Quotient law
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Same base, a ≠ 0. Subtract the indices.
Power of a power
(aᵐ)ⁿ = aᵐⁿ
Multiply the indices.
Power of a product
(ab)ⁿ = aⁿ × bⁿ
The index goes to each factor.
Power of a quotient
(a/b)ⁿ = aⁿ / bⁿ
For b ≠ 0.
Zero index
a⁰ = 1
For a ≠ 0. 0⁰ is not defined at this level.
Negative index
a⁻ⁿ = 1/aⁿ
For a ≠ 0. Also (a/b)⁻ⁿ = (b/a)ⁿ.
Fractional index
a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ
Take the root first, then the power.

How to solve Laws of Indices questions

Use this order for any question that asks you to simplify or evaluate an expression with indices.

  1. 1Write every number in its smallest prime base, for example 8 = 2³, 27 = 3³, 25 = 5².
  2. 2Remove negative indices by taking reciprocals. Move the term across the fraction line.
  3. 3Convert roots to fractional indices, for example √a = a^(1/2).
  4. 4Apply the power-of-a-power law and the power-of-a-product law to clear brackets.
  5. 5Group terms with the same base and add indices for multiplication and subtract for division.
  6. 6Simplify each index using fraction arithmetic. Watch signs carefully.
  7. 7Replace any zero index by 1, then evaluate and match with the options.

Quickest way: Common base and index check

When to use it: Use this when the options are numbers and the expression has several bases like 2, 4, 8 or 3, 9, 27.

  1. Convert all numbers to one prime base at once.
  2. Add up all indices in the numerator and all in the denominator separately.
  3. Subtract once at the end to get a single index.
  4. If the answer is a plain number, test it against the options by quick substitution of small values.

Common mistakes in Laws of Indices

  • Adding indices when bases are different, such as 2³ × 3² = 6⁵.

    Students remember the product law but forget it needs the same base.

    Fix: Check the bases first. If they differ, convert to a common base or evaluate separately: 8 × 9 = 72.

  • Treating a⁻ⁿ as a negative number, so 2⁻³ = −8.

    The minus sign in the index is confused with the sign of the number.

    Fix: A negative index means reciprocal: 2⁻³ = 1/2³ = 1/8.

  • Writing aᵐ + aⁿ = aᵐ⁺ⁿ.

    Students apply the product law to addition.

    Fix: The laws work only for multiplication and division. For addition, take out the common factor: 2³ + 2² = 2²(2 + 1) = 12.

  • Saying a⁰ = 0.

    Zero index is confused with zero value.

    Fix: Any non-zero number to the power 0 is 1. Also 3 × 5⁰ = 3, not 0.

  • Adding indices in (aᵐ)ⁿ to get aᵐ⁺ⁿ.

    Two laws with indices get mixed up.

    Fix: A power of a power means multiply: (2³)² = 2⁶ = 64. Product of powers means add: 2³ × 2² = 2⁵.

  • Applying the index only to the number in (2a)³ and writing 2a³.

    Brackets are ignored.

    Fix: The index applies to every factor inside: (2a)³ = 8a³.

Worked examples

Example 1

Simplify (2⁵ × 4³) ÷ 8² and find its value.

Show the solution
  1. Write 4 = 2² and 8 = 2³.
  2. 4³ = (2²)³ = 2⁶ and 8² = (2³)² = 2⁶.
  3. Numerator = 2⁵ × 2⁶ = 2¹¹.
  4. Divide: 2¹¹ ÷ 2⁶ = 2⁵.
  5. 2⁵ = 32.

Answer: 32

Example 2

Find the value of 27^(2/3) + 16^(−1/2) + 5⁰.

Show the solution
  1. 27^(2/3) = (27^(1/3))² = 3² = 9.
  2. 16^(−1/2) = 1/16^(1/2) = 1/4.
  3. 5⁰ = 1.
  4. Add: 9 + 1/4 + 1 = 10 + 1/4 = 41/4.

Answer: 41/4 (that is, 10.25)

Exam tips

  • Most questions are about converting to a common prime base. Memorise squares to 15 and cubes to 10, plus powers of 2, 3 and 5.
  • Read the sign of each index carefully. Options are often built from the common sign errors.
  • For fractional indices, take the root first. 64^(2/3) is easier as 4² than as the cube root of 4096.
  • Check whether the question says a ≠ 0 or gives a condition before using a⁰ = 1.
  • With 50 questions in one hour, spend under a minute here. If a question has unfamiliar bases, try substituting a simple value.

Practice questions from Indices and Logarithms

Laws of Indices in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Laws of Indices: frequently asked questions

What is the value of any number raised to the power zero?

It is 1, provided the number is not zero. This follows from aⁿ ÷ aⁿ = a⁰ and any non-zero number divided by itself is 1.

What does a negative index mean?

It means the reciprocal of the positive power. For example, 5⁻² = 1/5² = 1/25. The value stays positive for a positive base.

Can I use the laws of indices when the bases are different?

Add or subtract indices only when the bases are the same. If not, try to write both numbers with a common base, such as 4 and 8 as powers of 2. Otherwise evaluate each separately.

How do I solve fractional indices quickly?

Read the denominator as the root and the numerator as the power. For 8^(2/3), the cube root of 8 is 2 and 2² is 4. Doing the root first keeps numbers small.