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Index Laws

Indices are mathematical shorthand for repeated multiplication. Master the handful of rules on this page and you can simplify almost any algebraic expression at speed — a skill every later chapter relies on.

25 min Video by Zeeshan Zamurred Algebraic Expressions

Edexcel AS Level Maths: 1.1 Laws of IndicesWatch the full walkthrough before the notes below.

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What you'll be able to do

Use index notation and identify the base and the index (power)
Multiply and divide powers of the same base
Raise a power to another power and apply powers to products
Use the rule that any non-zero number to the power 0 equals 1
Combine several index laws in a single simplification

What an index actually means

An $index$ (plural: indices), also called a power or exponent, tells you how many times to multiply a number by itself. In $a^{n}$ , the number $a$ is the $base$ and $n$ is the $index$ .

So $2^{4} = 2 \times 2 \times 2 \times 2 = 16$ . Every index law below comes directly from this definition — if you ever forget a rule, write the power out in full and the pattern reappears.

a^{n} = n times a \times a \times \dots \times a

The base

a

is multiplied by itself

n

times.

Multiplying and dividing powers

When two powers share the same base, multiplying $adds$ the indices and dividing $subtracts$ them. This only works when the bases match.

a^{m} \times a^{n} = a^{m + n}

Same base, multiply ⟶ add the indices.

a^{m} \div a^{n} = a^{m - n}

Same base, divide ⟶ subtract the indices.

1The base

x

is the same, so add the indices.

5 + 3 = 8

Answer

x^{8}

Tip — The number in front (the coefficient) is multiplied or divided as normal — only the indices get added or subtracted.

Powers of powers and powers of products

Raising a power to another power $multiplies$ the indices. When a whole product is raised to a power, that power applies to every factor inside the bracket — including the coefficient.

(a^{m})^{n} = a^{mn}

Power of a power ⟶ multiply the indices.

(ab)^{n} = a^{n} b^{n}

The power spreads across each factor in the bracket.

1Multiply the indices:

4 \times 3 = 12

Answer

x^{12}

Tip — A classic slip is to forget the coefficient: $(3 x^{2})^{3}$ is $27 x^{6}$ , not $3 x^{6}$ . The 3 is cubed too.

The power of zero

Any non-zero number raised to the power $0$ equals $1$ . You can see why from the division law: $a^{n} \div a^{n} = a^{0}$ , but anything divided by itself is $1$ .

a^{0} = 1 (a \neq = 0)

For example

5^{0} = 1

and

(7 x y)^{0} = 1

1Only the

x

is raised to the power

0

, so

x^{0} = 1

2This leaves

4 \times 1

Answer

4

Combining the laws

Harder questions stack several laws together. Work from the inside of any brackets outwards, and keep coefficients separate from the variables until the end.

1Expand each bracket first using the power-of-a-product law.

(2 x^{2} y^{3})^{2} = 4 x^{4} y^{6}

(x y)^{3} = x^{3} y^{3}

4Now divide: numbers stay as

4

, then subtract indices for

x

(

4 - 3

) and

y

(

6 - 3

Answer

4 x y^{3}

Tip — Tidy each bracket on its own before you combine. Trying to do everything in one step is where errors creep in.

Formula recap

a^{m} \times a^{n} = a^{m + n}

Multiply powers ⟶ add indices.

a^{m} \div a^{n} = a^{m - n}

Divide powers ⟶ subtract indices.

(a^{m})^{n} = a^{mn}

Power of a power ⟶ multiply indices.

(ab)^{n} = a^{n} b^{n}, a^{0} = 1

Spread powers across products; power 0 gives 1.

Common mistakes to avoid

Adding indices with different bases, e.g. 2³ × 3⁴ = 6⁷.

Only add indices when the bases match. 2³ × 3⁴ stays as 8 × 81 = 648.

Forgetting to apply a bracket power to the coefficient: (2x)³ = 2x³.

Cube everything inside: (2x)³ = 2³x³ = 8x³.

Subtracting indices the wrong way round when dividing.

It is top index minus bottom index: x⁵ ÷ x² = x⁵⁻² = x³.

Key takeaways

Same base: multiply → add indices, divide → subtract indices.
Power of a power → multiply indices; a bracket power hits every factor, coefficients included.
Any non-zero number to the power 0 is 1.
For mixed problems, simplify each bracket first, then combine.

Test yourself

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