Indices

What you'll learn

What a base and an index mean.
How to multiply and divide powers with the same base.
How to simplify brackets such as $3x^2y)^3$ .
How zero indices, negative indices and reciprocals work.

The basics: powers, bases and indices

A power is a shorter way of writing repeated multiplication. For example, $4^3$ means:

43=4×4×44^3 = 4 \times 4 \times 4

It does not mean 4 times 3.

Definition

Base and index

In an expression such as $a^n$ , the base is $a$ and the index is $n$ . The plural of index is indices. The index tells you how many times the base is multiplied by itself.

Example

Evaluating a simple power

Expand $2^5$ as five factors of 2.
$25=2×2×2×2×22^5 = 2 \times 2 \times 2 \times 2 \times 2$
Multiply carefully from left to right.
$2^5 = 32$

Common Mistake

Index does not mean multiply

$2^5$ means 2 multiplied by itself five times. It is not the same as 2 times 5.

Multiplying powers with the same base

When you multiply powers with the same base, you add the indices.

Key Idea

Multiplication law

For the same base, $am×an=am+na^m \times a^n = a^{m+n}$ . The base stays the same; the indices add.

The diagram below shows the idea behind the three main laws: multiplying joins factors, dividing cancels factors, and a power outside brackets repeats the whole power.

Diagram showing the multiply, divide, and power index laws

Example

Simplifying a product of powers

Simplify $r7×r2r^7 \times r^2$ .
The base is the same in both terms: $r$ .
Add the indices.
$r7×r2=r7+2r^7 \times r^2 = r^{7+2}$
Write the answer as one power: $r^9$ .

Tip

Hidden index of 1

A letter on its own has index 1. So $x$ means $x^1$ .

Dividing powers with the same base

When you divide powers with the same base, you subtract the indices.

Key Idea

Division law

For the same non-zero base, $am÷an=am−na^m \div a^n = a^{m-n}$ . In fraction form, $aman=am−n\frac{a^m}{a^n} = a^{m-n}$ .

Example

Simplifying a quotient of powers

Simplify $y8y3\frac{y^8}{y^3}$ .
The base is the same: $y$ .
Subtract the bottom index from the top index.
$y8y3=y8−3\frac{y^8}{y^3} = y^{8-3}$
Write the answer: $y^5$ .

Common Mistake

Do not divide by zero

The division law assumes the base is not zero, because division by zero is not allowed.

Powers of powers

Sometimes a power is raised to another power, such as $x^3)^4$ .

This means the power $x^3$ is being repeated four times:

(x3)4=x3×x3×x3×x3(x^3)^4 = x^3 \times x^3 \times x^3 \times x^3

So the indices multiply.

Key Idea

Power of a power

When a power is raised to another power, multiply the indices: $a^m)^n = a^{mn}$ .

Example

Simplifying a power of a power

Simplify $t^4)^3$ .
Multiply the indices 4 and 3.
$(t4)3=t4×3(t^4)^3 = t^{4 \times 3}$
Write the final answer: $t^{12}$ .

Common Mistake

Do not add here

For $x^5)^2$ , multiply the indices to get $x^{10}$ . Do not add them to get $x^7$ .

Brackets with coefficients and letters

A bracketed expression may include a number as well as letters, such as $2a^3b)^4$ .

Definition

Coefficient

A coefficient is the number multiplying a letter. In $6 x$ , the coefficient of $x$ is 6.

When a bracket is raised to a power, the outside index applies to everything inside the bracket.

Example

Simplifying brackets with a coefficient

Simplify $3m^2n)^3$ .
Cube the coefficient 3.
$3^3 = 27$
Multiply each letter’s index by 3.
$m^2)^3 = m^6$
Remember that $n$ has a hidden index of 1, so $n^3$ stays as $n^3$ .
Combine the parts: $27m^6n^3$ .

Common Mistake

Forgetting to power the number

In $4x)^2$ , the answer is $16x^2$ , not $4x^2$ , because the coefficient 4 is squared as well.

Multiplying algebraic terms

For expressions with coefficients and letters, deal with the numbers and each letter separately.

Key Idea

Multiply term by term

Multiply the coefficients, then add indices for matching letters.

Example

Multiplying terms with different letters

Simplify $4p2q3×5pq24p^2q^3 \times 5pq^2$ .
Multiply the coefficients.
$\times 5 = 20$
Add the powers of $p$ , remembering $p=p^1$ .
$p2×p=p2+1=p3p^2 \times p = p^{2+1} = p^3$
Add the powers of $q$ .
$q3×q2=q3+2=q5q^3 \times q^2 = q^{3+2} = q^5$
Combine everything: $20p^3q^5$ .

Dividing algebraic terms

When dividing algebraic terms, divide the coefficients and subtract indices for matching letters.

Example

Dividing algebraic terms

Simplify $18a6b46a2b\frac{18a^6b^4}{6a^2b}$ .
Divide the coefficients.
$18 \div 6 = 3$
Subtract the powers of $a$ .
$a6÷a2=a6−2=a4a^6 \div a^2 = a^{6-2} = a^4$
Subtract the powers of $b$ , remembering $b=b^1$ .
$b4÷b=b4−1=b3b^4 \div b = b^{4-1} = b^3$
Combine everything: $3a^4b^3$ .

Tip

Cancel first if it helps

In a fraction, you can cancel common factors before using index laws. This is often useful when the coefficients are not simple multiples.

Zero indices

Any non-zero number or letter to the power of 0 is equal to 1.

Why? Because dividing a power by itself gives 1:

x4x4=1\frac{x^4}{x^4}=1

Using index laws:

x4x4=x4−4=x0\frac{x^4}{x^4}=x^{4-4}=x^0

So $x^0=1$ when $\neq 0$ .

Key Idea

Zero index

For any non-zero base, $a^0=1$ .

Example

Using a zero index

Find the value of $9^0$ .
The base 9 is not zero.
Use the zero index rule.
$9^0 = 1$

Negative indices and reciprocals

A negative index means “take the reciprocal”.

Definition

Reciprocal

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 5 is $15\frac{1}{5}$ , and the reciprocal of $14\frac{1}{4}$ is 4.

Key Idea

Negative index

For a non-zero base, $a−n=1ana^{-n}=\frac{1}{a^n}$ .

Example

Evaluating a negative power

Work out $3^{-2}$ .
A negative index means write the reciprocal.
$3−2=1323^{-2} = \frac{1}{3^2}$
Evaluate the positive power in the denominator.
$3−2=193^{-2} = \frac{1}{9}$

Example

Simplifying with a negative index

Simplify $7x5×2x−37x^5 \times 2x^{-3}$ .
Multiply the coefficients.
$\times 2 = 14$
Add the indices of $x$ .
$x5×x−3=x5+(−3)=x2x^5 \times x^{-3} = x^{5+(-3)} = x^2$
Combine the parts: $14x^2$ .

Rewriting numbers as powers

Sometimes you need to rewrite a number using the same base as the rest of the expression.

For example, 25 can be written as $5^2$ , and 1000 can be written as $10^3$ .

Example

Writing as a power of the same base

Write $25 \times 5^4$ as a power of 5.
Rewrite 25 as a power of 5.
$25 = 5^2$
Substitute this into the expression.
$25 \times 5^4 = 5^2 \times 5^4$
Add the indices.
$52×54=565^2 \times 5^4 = 5^6$

Example

Changing a power of 1000 into a power of 10

Write $1000^3$ as a power of 10.
Rewrite 1000 as $10^3$ .
Put this into the expression.
$1000^3 = (10^3)^3$
Multiply the indices.
$10^3)^3 = 10^9$

Finding missing indices

If an index is unknown, use the index law first, then solve the simple equation.

Example

Finding missing powers

Find $k$ if $z4×zk=z11z^4 \times z^k = z^{11}$ .
Multiplying same bases means add the indices.
$z^{4+k}=z^{11}$
Equate the indices.
$4 + k = 11$
Solve to get $k = 7$ .

Example

A missing power in brackets

Find $b$ if $h^3)^b=h^{15}$ .
A power of a power means multiply the indices.
$h^3)^b = h^{3b}$
Equate the indices.
$3 b = 15$
Solve to get $b = 5$ .

Example

A missing power in a division

Find $a$ if $x5xa=x8\frac{x^5}{x^a}=x^8$ .
Dividing same bases means subtract the indices.
$x5xa=x5−a\frac{x^5}{x^a}=x^{5-a}$
Equate the indices.
$5 - a = 8$
Solve to get $a = - 3$ .

Combining index laws before evaluating

In numerical questions, it is often quicker to simplify the powers first, then work out the final value.

Example

Working out a value using index laws

Work out $27×2−223\frac{2^7 \times 2^{-2}}{2^3}$ .
Combine the powers of 2 by adding the top indices and subtracting the bottom index.
$27×2−223=27+(−2)−3\frac{2^7 \times 2^{-2}}{2^3}=2^{7+(-2)-3}$
Simplify the index.
$2^{7+(-2)-3}=2^2$
Evaluate the final power.
$2^2=4$

Exam technique

In the exam

Check the bases are the same before adding or subtracting indices.
When there are brackets, apply the outside index to the coefficient and every letter inside.
Remember the special rules: $a^0=1$ and $a−n=1ana^{-n}=\frac{1}{a^n}$ for non-zero $a$ .

Self review

Check yourself

Can you explain the difference between $5^3$ and 5 times 3?
Can you simplify an expression with coefficients, such as $6x4y2÷3x2y6x^4y^2 \div 3x^2y$ ?
Can you rewrite a negative index as a reciprocal?

Indices

What you'll learn

What a base and an index mean.
How to multiply and divide powers with the same base.
How to simplify brackets such as $3x^2y)^3$ .
How zero indices, negative indices and reciprocals work.

The basics: powers, bases and indices

A power is a shorter way of writing repeated multiplication. For example, $4^3$ means:

43=4×4×44^3 = 4 \times 4 \times 4

It does not mean 4 times 3.

Definition

Base and index

In an expression such as $a^n$ , the base is $a$ and the index is $n$ . The plural of index is indices. The index tells you how many times the base is multiplied by itself.

Example

Evaluating a simple power

Expand $2^5$ as five factors of 2.
$25=2×2×2×2×22^5 = 2 \times 2 \times 2 \times 2 \times 2$
Multiply carefully from left to right.
$2^5 = 32$

Common Mistake

Index does not mean multiply

$2^5$ means 2 multiplied by itself five times. It is not the same as 2 times 5.

Multiplying powers with the same base

When you multiply powers with the same base, you add the indices.

Key Idea

Multiplication law

For the same base, $am×an=am+na^m \times a^n = a^{m+n}$ . The base stays the same; the indices add.

The diagram below shows the idea behind the three main laws: multiplying joins factors, dividing cancels factors, and a power outside brackets repeats the whole power.

Diagram showing the multiply, divide, and power index laws

Example

Simplifying a product of powers

Simplify $r7×r2r^7 \times r^2$ .
The base is the same in both terms: $r$ .
Add the indices.
$r7×r2=r7+2r^7 \times r^2 = r^{7+2}$
Write the answer as one power: $r^9$ .

Tip

Hidden index of 1

A letter on its own has index 1. So $x$ means $x^1$ .

Dividing powers with the same base

When you divide powers with the same base, you subtract the indices.

Key Idea

Division law

For the same non-zero base, $am÷an=am−na^m \div a^n = a^{m-n}$ . In fraction form, $aman=am−n\frac{a^m}{a^n} = a^{m-n}$ .

Example

Simplifying a quotient of powers

Simplify $y8y3\frac{y^8}{y^3}$ .
The base is the same: $y$ .
Subtract the bottom index from the top index.
$y8y3=y8−3\frac{y^8}{y^3} = y^{8-3}$
Write the answer: $y^5$ .

Common Mistake

Do not divide by zero

The division law assumes the base is not zero, because division by zero is not allowed.

Powers of powers

Sometimes a power is raised to another power, such as $x^3)^4$ .

This means the power $x^3$ is being repeated four times:

(x3)4=x3×x3×x3×x3(x^3)^4 = x^3 \times x^3 \times x^3 \times x^3

So the indices multiply.

Key Idea

Power of a power

When a power is raised to another power, multiply the indices: $a^m)^n = a^{mn}$ .

Example

Simplifying a power of a power

Simplify $t^4)^3$ .
Multiply the indices 4 and 3.
$(t4)3=t4×3(t^4)^3 = t^{4 \times 3}$
Write the final answer: $t^{12}$ .

Common Mistake

Do not add here

For $x^5)^2$ , multiply the indices to get $x^{10}$ . Do not add them to get $x^7$ .

Brackets with coefficients and letters

A bracketed expression may include a number as well as letters, such as $2a^3b)^4$ .

Definition

Coefficient

A coefficient is the number multiplying a letter. In $6 x$ , the coefficient of $x$ is 6.

When a bracket is raised to a power, the outside index applies to everything inside the bracket.

Example

Simplifying brackets with a coefficient

Simplify $3m^2n)^3$ .
Cube the coefficient 3.
$3^3 = 27$
Multiply each letter’s index by 3.
$m^2)^3 = m^6$
Remember that $n$ has a hidden index of 1, so $n^3$ stays as $n^3$ .
Combine the parts: $27m^6n^3$ .

Common Mistake

Forgetting to power the number

In $4x)^2$ , the answer is $16x^2$ , not $4x^2$ , because the coefficient 4 is squared as well.

Multiplying algebraic terms

For expressions with coefficients and letters, deal with the numbers and each letter separately.

Key Idea

Multiply term by term

Multiply the coefficients, then add indices for matching letters.

Example

Multiplying terms with different letters

Simplify $4p2q3×5pq24p^2q^3 \times 5pq^2$ .
Multiply the coefficients.
$\times 5 = 20$
Add the powers of $p$ , remembering $p=p^1$ .
$p2×p=p2+1=p3p^2 \times p = p^{2+1} = p^3$
Add the powers of $q$ .
$q3×q2=q3+2=q5q^3 \times q^2 = q^{3+2} = q^5$
Combine everything: $20p^3q^5$ .

Dividing algebraic terms

When dividing algebraic terms, divide the coefficients and subtract indices for matching letters.

Example

Dividing algebraic terms

Simplify $18a6b46a2b\frac{18a^6b^4}{6a^2b}$ .
Divide the coefficients.
$18 \div 6 = 3$
Subtract the powers of $a$ .
$a6÷a2=a6−2=a4a^6 \div a^2 = a^{6-2} = a^4$
Subtract the powers of $b$ , remembering $b=b^1$ .
$b4÷b=b4−1=b3b^4 \div b = b^{4-1} = b^3$
Combine everything: $3a^4b^3$ .

Tip

Cancel first if it helps

In a fraction, you can cancel common factors before using index laws. This is often useful when the coefficients are not simple multiples.

Zero indices

Any non-zero number or letter to the power of 0 is equal to 1.

Why? Because dividing a power by itself gives 1:

x4x4=1\frac{x^4}{x^4}=1

Using index laws:

x4x4=x4−4=x0\frac{x^4}{x^4}=x^{4-4}=x^0

So $x^0=1$ when $\neq 0$ .

Key Idea

Zero index

For any non-zero base, $a^0=1$ .

Example

Using a zero index

Find the value of $9^0$ .
The base 9 is not zero.
Use the zero index rule.
$9^0 = 1$

Negative indices and reciprocals

A negative index means “take the reciprocal”.

Definition

Reciprocal

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 5 is $15\frac{1}{5}$ , and the reciprocal of $14\frac{1}{4}$ is 4.

Key Idea

Negative index

For a non-zero base, $a−n=1ana^{-n}=\frac{1}{a^n}$ .

Example

Evaluating a negative power

Work out $3^{-2}$ .
A negative index means write the reciprocal.
$3−2=1323^{-2} = \frac{1}{3^2}$
Evaluate the positive power in the denominator.
$3−2=193^{-2} = \frac{1}{9}$

Example

Simplifying with a negative index

Simplify $7x5×2x−37x^5 \times 2x^{-3}$ .
Multiply the coefficients.
$\times 2 = 14$
Add the indices of $x$ .
$x5×x−3=x5+(−3)=x2x^5 \times x^{-3} = x^{5+(-3)} = x^2$
Combine the parts: $14x^2$ .

Rewriting numbers as powers

Sometimes you need to rewrite a number using the same base as the rest of the expression.

For example, 25 can be written as $5^2$ , and 1000 can be written as $10^3$ .

Example

Writing as a power of the same base

Write $25 \times 5^4$ as a power of 5.
Rewrite 25 as a power of 5.
$25 = 5^2$
Substitute this into the expression.
$25 \times 5^4 = 5^2 \times 5^4$
Add the indices.
$52×54=565^2 \times 5^4 = 5^6$

Example

Changing a power of 1000 into a power of 10

Write $1000^3$ as a power of 10.
Rewrite 1000 as $10^3$ .
Put this into the expression.
$1000^3 = (10^3)^3$
Multiply the indices.
$10^3)^3 = 10^9$

Finding missing indices

If an index is unknown, use the index law first, then solve the simple equation.

Example

Finding missing powers

Find $k$ if $z4×zk=z11z^4 \times z^k = z^{11}$ .
Multiplying same bases means add the indices.
$z^{4+k}=z^{11}$
Equate the indices.
$4 + k = 11$
Solve to get $k = 7$ .

Example

A missing power in brackets

Find $b$ if $h^3)^b=h^{15}$ .
A power of a power means multiply the indices.
$h^3)^b = h^{3b}$
Equate the indices.
$3 b = 15$
Solve to get $b = 5$ .

Example

A missing power in a division

Find $a$ if $x5xa=x8\frac{x^5}{x^a}=x^8$ .
Dividing same bases means subtract the indices.
$x5xa=x5−a\frac{x^5}{x^a}=x^{5-a}$
Equate the indices.
$5 - a = 8$
Solve to get $a = - 3$ .

Combining index laws before evaluating

In numerical questions, it is often quicker to simplify the powers first, then work out the final value.

Example

Working out a value using index laws

Work out $27×2−223\frac{2^7 \times 2^{-2}}{2^3}$ .
Combine the powers of 2 by adding the top indices and subtracting the bottom index.
$27×2−223=27+(−2)−3\frac{2^7 \times 2^{-2}}{2^3}=2^{7+(-2)-3}$
Simplify the index.
$2^{7+(-2)-3}=2^2$
Evaluate the final power.
$2^2=4$

Exam technique

In the exam

Check the bases are the same before adding or subtracting indices.
When there are brackets, apply the outside index to the coefficient and every letter inside.
Remember the special rules: $a^0=1$ and $a−n=1ana^{-n}=\frac{1}{a^n}$ for non-zero $a$ .

Self review

Check yourself

Can you explain the difference between $5^3$ and 5 times 3?
Can you simplify an expression with coefficients, such as $6x4y2÷3x2y6x^4y^2 \div 3x^2y$ ?
Can you rewrite a negative index as a reciprocal?

Indices

What you'll learn

The basics: powers, bases and indices

Multiplying powers with the same base

Dividing powers with the same base

Powers of powers

Brackets with coefficients and letters

Multiplying algebraic terms

Dividing algebraic terms

Zero indices

Negative indices and reciprocals

Rewriting numbers as powers

Finding missing indices

Combining index laws before evaluating

Recap questions

You've reached the end

Indices

What you'll learn

The basics: powers, bases and indices

Multiplying powers with the same base

Dividing powers with the same base

Powers of powers

Brackets with coefficients and letters

Multiplying algebraic terms

Dividing algebraic terms

Zero indices

Negative indices and reciprocals

Rewriting numbers as powers

Finding missing indices

Combining index laws before evaluating

Recap questions

You've reached the end

Indices

What you'll learn

The basics: powers, bases and indices

Multiplying powers with the same base

Dividing powers with the same base

Powers of powers

Brackets with coefficients and letters

Multiplying algebraic terms

Dividing algebraic terms

Zero indices

Negative indices and reciprocals

Rewriting numbers as powers

Finding missing indices

Combining index laws before evaluating

Recap questions

You've reached the end

Indices Revision Guide

Indices

What you'll learn

The basics: powers, bases and indices

Multiplying powers with the same base

Dividing powers with the same base

Powers of powers

Brackets with coefficients and letters

Multiplying algebraic terms

Dividing algebraic terms

Zero indices

Negative indices and reciprocals

Rewriting numbers as powers

Finding missing indices

Combining index laws before evaluating

Recap questions

You've reached the end

Indices Revision Guide