Laws of Indices

Welcome to Laws of Indices!

An index (or exponent) tells you how many times a base is multiplied by itself. The Laws of Indices are a set of rules that let you simplify, expand, and solve expressions involving powers quickly and reliably. They underpin algebra, equations, surds, and much of IGCSE mathematics.

a^m means a × a × a × … (m times) | Base: a | Index: m

There are 9 key laws to master. Each one has a simple justification based on what multiplication really means. Once you understand the why, the rules become easy to remember and apply.

Multiplication Law

a^m × aⁿ = a^m+n

Division Law

a^m ÷ aⁿ = a^m−n

Power Law

(a^m)ⁿ = a^mn

Zero Index

a⁰ = 1

Negative Index

a⁻ⁿ = 1/aⁿ

Fractional Index

a^1/n = ⁿ√a

General Fraction

a^m/n = (ⁿ√a)^m

Bracket Laws

(ab)ⁿ and (a/b)ⁿ

Solving Equations

2^x = 32 → x = ?

The 9 Laws — Quick Reference

Law	Rule	Example
Multiplication	a^m × aⁿ = a^m+n	2³ × 2⁴ = 2⁷ = 128
Division	a^m ÷ aⁿ = a^m−n	3⁵ ÷ 3² = 3³ = 27
Power of Power	(a^m)ⁿ = a^mn	(2³)⁴ = 2¹²
Zero Index	a⁰ = 1 (a ≠ 0)	7⁰ = 1, (−3)⁰ = 1
Negative Index	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8
Fractional Index (root)	a^1/n = ⁿ√a	8^1/3 = ∛8 = 2
Fractional Index (general)	a^m/n = (ⁿ√a)^m	8^2/3 = (∛8)² = 4
Bracket — product	(ab)ⁿ = aⁿbⁿ	(2×3)² = 4×9 = 36
Bracket — quotient	(a/b)ⁿ = aⁿ/bⁿ	(2/3)³ = 8/27

1. Multiplication Law: a^m × aⁿ = a^m+n

Why it works: a^m means m copies of a multiplied together, and aⁿ means n more copies. Put them together and you have m + n copies total.

Proof:
a³ × a⁴ = (a × a × a) × (a × a × a × a) = a⁷ ✓
Counting: 3 + 4 = 7 factors of a.

a^m × aⁿ = a^m+n — add the indices when multiplying same base

Examples:
5² × 5³ = 5⁵ = 3125
x⁴ × x⁷ = x¹¹
3a² × 5a³ = 15a⁵ (multiply coefficients, add indices)
2x³y² × 4x²y = 8x⁵y³

Only add indices when the bases are the same. x² × y³ cannot be simplified further — the bases differ.

2. Division Law: a^m ÷ aⁿ = a^m−n

Why it works: Division cancels pairs of factors. Each a in the denominator cancels one a in the numerator, leaving m − n factors of a.

Proof:
a⁵ ÷ a² = (a×a×a×a×a) / (a×a) = a³ ✓
5 − 2 = 3 factors remain.

a^m ÷ aⁿ = a^m−n — subtract the indices when dividing same base

Examples:
2⁷ ÷ 2³ = 2⁴ = 16
x⁹ ÷ x⁴ = x⁵
12x⁵ ÷ 4x² = 3x³ (divide coefficients, subtract indices)
6a⁴b³ ÷ 2a²b = 3a²b²

3. Power of a Power: (a^m)ⁿ = a^mn

Why it works: Raising a^m to the power n means multiplying a^m by itself n times. By the multiplication law, you add m repeatedly n times: m + m + … + m = mn.

Proof:
(a³)⁴ = a³ × a³ × a³ × a³ = a^3+3+3+3 = a¹² ✓
Alternatively: 3 × 4 = 12.

(a^m)ⁿ = a^mn — multiply the indices for a power of a power

Examples:
(5²)³ = 5⁶ = 15625
(x⁴)⁵ = x²⁰
(2x³)⁴ = 2⁴x¹² = 16x¹²
(3a²b)³ = 27a⁶b³

When a bracket contains a coefficient, raise the coefficient to that power too: (2x³)⁴ = 2⁴ × x^3×4 = 16x¹².

4. Zero Index: a⁰ = 1

Why it works: Use the division law. aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But any number divided by itself is 1. Therefore a⁰ = 1.

a⁰ = 1 for any non-zero value of a

7⁰ = 1 100⁰ = 1 (−5)⁰ = 1
(x²y³)⁰ = 1 3x⁰ = 3×1 = 3 (only x is raised to 0)

Watch out: 3x⁰ = 3 because only x is raised to the power 0, not the whole term 3x. If the whole term is raised to 0, write it as (3x)⁰ = 1.

5. Negative Index: a⁻ⁿ = 1/aⁿ

Why it works: Using the division law: a² ÷ a⁵ = a²⁻⁵ = a⁻³. But also a²/a⁵ = 1/a³. So a⁻³ = 1/a³.

a⁻ⁿ = 1/aⁿ — a negative index means "take the reciprocal"

Examples:
2⁻³ = 1/2³ = 1/8
3⁻² = 1/9
x⁻⁴ = 1/x⁴
(1/2)⁻³ = 2³ = 8 (reciprocal of 1/2 is 2)
5x⁻² = 5/x²

To evaluate a⁻ⁿ with a fraction base: (a/b)⁻ⁿ = (b/a)ⁿ. The negative index flips the fraction.

6. Fractional Index (Root): a^1/n = ⁿ√a

Why it works: If we want a^1/n to obey the power law, then (a^1/n)ⁿ = a¹ = a. So a^1/n must be the number that gives a when raised to the power n — that is the nth root of a.

a^1/n = ⁿ√a — a fractional index of 1/n is the nth root

Examples:
9^1/2 = √9 = 3
8^1/3 = ∛8 = 2
16^1/4 = ⁴√16 = 2
64^1/2 = 8 27^1/3 = 3 32^1/5 = 2

Connection to surds: 2^1/2 = √2. Fractional indices with denominator 2 are the same as square roots — surds and indices are the same idea expressed differently.

7. General Fractional Index: a^m/n = (ⁿ√a)^m

Why it works: a^m/n = (a^1/n)^m = (ⁿ√a)^m by the power law combined with the root law. You can also write it as ⁿ√(a^m), but root first then power is usually easier to compute.

a^m/n = (ⁿ√a)^m — root first (denominator), then power (numerator)

Examples:
8^2/3 = (∛8)² = 2² = 4
27^2/3 = (∛27)² = 3² = 9
16^3/4 = (⁴√16)³ = 2³ = 8
4^3/2 = (√4)³ = 2³ = 8
32^3/5 = (⁵√32)³ = 2³ = 8

With negative fractional index:
8^−2/3 = 1/8^2/3 = 1/(∛8)² = 1/4
27^−1/3 = 1/∛27 = 1/3

Memory tip: m/n → denominator is the Root, numerator is the Power. Think D.R. → R.P. → Root then Power.

8. Bracket Laws: (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ

When a bracket contains a product or quotient and is raised to a power, distribute the index to every factor inside.

(ab)ⁿ = aⁿbⁿ (a/b)ⁿ = aⁿ/bⁿ

Proof of (ab)ⁿ:
(ab)³ = (ab)(ab)(ab) = a·a·a · b·b·b = a³b³ ✓

Examples:
(3x)² = 9x²
(2xy)³ = 8x³y³
(x/2)⁴ = x⁴/16
(2a²b)³ = 8a⁶b³

9. Solving Index Equations: 2^x = 32

When solving equations like 2^x = 32, rewrite both sides with the same base, then equate the indices.

If a^x = a^k then x = k — same base means indices are equal

Example 1: Solve 2^x = 32
32 = 2⁵, so 2^x = 2⁵ → x = 5

Example 2: Solve 3^x = 1/9
1/9 = 1/3² = 3⁻², so 3^x = 3⁻² → x = −2

Example 3: Solve 4^x = 8
Write both as powers of 2: 4 = 2², 8 = 2³
(2²)^x = 2³ → 2^2x = 2³ → 2x = 3 → x = 3/2

Example 4: Solve 5^2x−1 = 125
125 = 5³, so 2x − 1 = 3 → 2x = 4 → x = 2

Key bases to know: 2, 4, 8, 16, 32, 64 (powers of 2); 3, 9, 27, 81 (powers of 3); 5, 25, 125 (powers of 5); 2, 8, 32 = 2¹,2³,2⁵; 4, 16, 64 = 2²,2⁴,2⁶.

10. Simplifying Algebraic Expressions

Apply multiple laws in sequence to simplify complex algebraic expressions with indices.

Example 1: Simplify (2x³y²)³ ÷ (4x²y)
= 8x⁹y⁶ ÷ 4x²y [bracket law]
= 2x⁷y⁵ [division law]

Example 2: Simplify 3a²b³ × 2a⁻¹b²
= 6a²⁺⁽⁻¹⁾b³⁺²
= 6a¹b⁵ = 6ab⁵

Work systematically: deal with coefficients first, then each variable's index separately using the relevant law.

Example 1 — Evaluate 27^2/3

Identify: fractional index m/n = 2/3. Root = 3 (cube root), Power = 2.

Root first: ∛27 = 3 (since 3³ = 27)

Then power: 3² = 9

Answer: 27^2/3 = 9

Example 2 — Evaluate 16^−3/4

Negative index: 16^−3/4 = 1 / 16^3/4

Root first: ⁴√16 = 2 (since 2⁴ = 16)

Power: 2³ = 8, so 16^3/4 = 8

Negative: 1/8. Answer: 16^−3/4 = 1/8

Example 3 — Simplify (3x²y)³

Apply bracket law: raise each factor to the power 3

3³ = 27 (x²)³ = x⁶ y³ = y³

Answer: (3x²y)³ = 27x⁶y³

Example 4 — Simplify 4x³y⁵ × 3x²y⁻²

Coefficients: 4 × 3 = 12

x terms: x³ × x² = x³⁺² = x⁵

y terms: y⁵ × y⁻² = y⁵⁺⁽⁻²⁾ = y³

Answer: 12x⁵y³

Example 5 — Simplify (8x⁶) ÷ (2x⁴)²

Simplify denominator first: (2x⁴)² = 4x⁸

Divide: 8x⁶ ÷ 4x⁸ = 2x⁶⁻⁸ = 2x⁻²

Write without negative index: 2/x²

Answer: 2x⁻² = 2/x²

Example 6 — Solve 9^x = 27

Common base: 9 = 3² and 27 = 3³

Rewrite: (3²)^x = 3³ → 3^2x = 3³

Equate indices: 2x = 3

Answer: x = 3/2

Multiplication Law Visualiser: a^m × aⁿ = a^m+n

Choose a base and two indices. See the individual groups of factors and their combined count visually.

Base a = m = n =

Choose values and click Visualise.

Index Law Calculator

Evaluate any expression of the form a^m/n step by step.

Base a = m = n = Negative (a^−m/n)

Result will appear here.

Index Equation Solver

Solve a^x = c by rewriting both sides in the same base.

Base a = c =

Result will appear here.

Exercise 1 — Multiplication & Division Law (Numbers)

Use aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Give the index of the simplified answer (the power, not the full value).

1. 2³ × 2⁴ = 2^? Give the index.

2. 3² × 3⁵ = 3^? Give the index.

3. 5⁶ ÷ 5² = 5^? Give the index.

4. 7⁸ ÷ 7³ = 7^? Give the index.

5. 4¹ × 4⁶ = 4^? Give the index.

6. 2¹⁰ ÷ 2⁶ = 2^? Give the index.

Exercise 2 — Power of a Power & Zero Index

Use (aᵐ)ⁿ = aᵐⁿ and a⁰ = 1. Give the index of the simplified answer or the integer value.

1. (2³)⁴ = 2^? Give the index.

2. (5²)³ = 5^? Give the index.

3. (3⁴)² = 3^? Give the index.

4. 9⁰ = ? Give the value.

5. (−7)⁰ = ? Give the value.

6. (x⁵)⁴ = x^? Give the index.

Exercise 3 — Negative Indices (Evaluate)

Use a⁻ⁿ = 1/aⁿ. Give the answer as a fraction (numerator only, since denominator is given) or as a decimal where indicated.

1. 2⁻¹ = 1/? Give the denominator.

2. 2⁻³ = 1/? Give the denominator.

3. 3⁻² = 1/? Give the denominator.

4. 5⁻² = 1/? Give the denominator.

5. 4⁻¹ = 1/? Give the denominator.

6. 10⁻³ = 1/? Give the denominator.

Exercise 4 — Fractional Indices (Evaluate)

Use a^1/n = ⁿ√a and a^m/n = (ⁿ√a)^m. Give the integer value.

1. 9^1/2 = ? Give the value.

2. 27^1/3 = ? Give the value.

3. 16^1/4 = ? Give the value.

4. 8^2/3 = ? Give the value.

5. 4^3/2 = ? Give the value.

6. 32^2/5 = ? Give the value.

Exercise 5 — Solving Index Equations

Solve each equation by writing both sides with the same base. Give the value of x (may be a fraction — enter as a decimal).

1. 2^x = 8 → x = ?

2. 2^x = 32 → x = ?

3. 3^x = 27 → x = ?

4. 3^x = 1/9 → x = ?

5. 5^x = 125 → x = ?

6. 4^x = 8 → x = ? (enter as decimal)

Practice — 20 Questions

Mixed practice covering all Laws of Indices. Read each question carefully and enter the correct value.

1. 2⁴ × 2³ = 2^? Give the index.

2. 5⁷ ÷ 5³ = 5^? Give the index.

3. (3²)⁵ = 3^? Give the index.

4. 12⁰ = ? Give the value.

5. 2⁻⁴ = 1/? Give the denominator.

6. 25^1/2 = ? Give the value.

7. 64^1/3 = ? Give the value.

8. 27^2/3 = ? Give the value.

9. 16^3/4 = ? Give the value.

10. Solve 2^x = 64. x = ?

11. Solve 3^x = 81. x = ?

12. Solve 5^x = 1/25. x = ?

13. 3⁵ × 3⁻² = 3^? Give the index.

14. (2⁴)³ ÷ 2⁸ = 2^? Give the index.

15. 8^−1/3 = 1/? Give the denominator.

16. (1/3)⁻² = ? Give the integer value.

17. 32^3/5 = ? Give the value.

18. 4^5/2 = ? Give the value.

19. Solve 4^x = 32 (write x as a decimal).

20. Solve 9^x = 27 (write x as a decimal).

Challenge — 8 Questions

Harder questions involving multiple laws, algebraic expressions, and problem solving. Give numerical answers or index values as instructed.

1. Evaluate 125^2/3. Give the integer value.

2. Evaluate 81^−3/4 = 1/? Give the denominator.

3. Solve 8^x = 2. Give x as a fraction (decimal). Hint: write 8 = 2³.

4. Solve 25^x = 125. Give x as a decimal. Hint: use base 5.

5. Simplify (4x³)² ÷ (2x⁴). Result = k·x^p. Give the value of p (the index of x).

6. If a³ = 8, what is a⁻²? Give the answer as a fraction: 1/? Enter the denominator.

7. Evaluate (27/8)^2/3. Result = p/q. Enter the numerator p.

8. Solve 2^3x−1 = 16. Give the value of x.