Indices

What you'll learn

What an index means and how to read powers like $x^5$
How to multiply and divide powers with the same base
How to simplify brackets such as $4a^2b)^3$
How zero powers, negative powers and reciprocals work

1. What is an index?

An index tells you how many times a number or letter is multiplied by itself.

In $5^3$ , the 5 is the base and the 3 is the index. You read it as “5 to the power of 3”.

The base is the number being multiplied, and the index shows how many copies of it are used.

Definition

Base and index

In $a^n$ , the base is $a$ and the index is $n$ . It means multiply $a$ by itself $n$ times.

So:

53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125

Example

Working out powers

Work out the value of $42×334^2 \times 3^3$ .

Each power is expanded first, then the two results are multiplied.

Work out each power separately: $4^2 = 16$ and $3^3 = 27$ .
Multiply the results: 16 × 27 = 432.
So $42×33=4324^2 \times 3^3 = 432$ .

2. Multiplying powers with the same base

When the base is the same, you add the indices.

Key Idea

Multiplying same-base powers

am×an=am+na^m \times a^n = a^{m+n}

For example, $x4×x2x^4 \times x^2$ means four lots of $x$ multiplied by two more lots of $x$ , making six lots altogether.

Multiplying same-base powers combines all the repeated factors, so the indices add.

Example

Simplifying a product

Simplify $x6×x5x^6 \times x^5$ .

Check the bases are the same. Both powers have base $x$ .
Add the indices: 6 + 5 = 11.
Write the answer as one power: $x^{11}$ .

Common Mistake

Do not multiply the indices

For $x6×x5x^6 \times x^5$ , the answer is $x^{11}$ , not $x^{30}$ . Multiplying powers with the same base means add the indices.

3. Dividing powers with the same base

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

Key Idea

Dividing same-base powers

aman=am−n\frac{a^m}{a^n} = a^{m-n}

This works when $\neq 0$ .

Example

Simplifying a division

Simplify $w9w4\frac{w^9}{w^4}$ .

Dividing powers with the same base cancels matching factors from the numerator and denominator.

Check the bases are the same. Both powers have base $w$ .
Subtract the indices: 9 − 4 = 5.
Write the answer as one power: $w^5$ .

You may also see division written with the divide symbol, such as $d8÷d3d^8 \div d^3$ . It means the same as a fraction.

Example

Dividing using the divide sign

Simplify $d10÷d6d^{10} \div d^6$ .

The base is the same, so subtract the indices.
Work out 10 − 6 = 4.
The simplified answer is $d^4$ .

4. Coefficients and letters together

A coefficient is the number in front of a letter. For example, in $7p^3$ , the coefficient is 7.

When simplifying expressions with numbers and letters, deal with the numbers first, then deal with each letter using index laws.

Example

Multiplying numbers and letters

Simplify $4m3n2×6mn54m^3n^2 \times 6mn^5$ .

Separate the coefficients and each letter base, then add indices only for matching letters.

Multiply the coefficients: 4 × 6 = 24.
For the $m$ terms, remember that $m$ means $m^1$ . So $m3×m=m4m^3 \times m = m^4$ .
For the $n$ terms, add the indices: $n2×n5=n7n^2 \times n^5 = n^7$ .
Put everything together: $24m^4n^7$ .

Tip

Invisible power

A letter on its own has an invisible index of 1. So $m$ means $m^1$ , and $q$ means $q^1$ .

A single letter has an understood power of 1, which is useful when applying index laws.

Example

Dividing numbers and letters

Simplify $18p5q46p2q\frac{18p^5q^4}{6p^2q}$ .

Divide the coefficients: 18 ÷ 6 = 3.
Subtract the indices for $p$ : $p5÷p2=p3p^5 \div p^2 = p^3$ .
Subtract the indices for $q$ : $q4÷q=q3q^4 \div q = q^3$ .
Put everything together: $3p^3q^3$ .

5. Powers outside brackets

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

The outside power acts on every factor inside the brackets, including any coefficient.

Key Idea

Power of a power

(am)n=amn\left(a^m\right)^n = a^{mn}

So you multiply the indices.

Example

Power of a power

Simplify $(t4)3\left(t^4\right)^3$ .

The base is $t$ , and there is a power raised to another power.
Multiply the indices: 4 × 3 = 12.
The answer is $t^{12}$ .

If there is a number and more than one letter inside the bracket, raise every part to the outside power.

Example

Brackets with numbers and letters

Simplify $(3a2b)3\left(3a^2b\right)^3$ .

Cube the coefficient: $3^3 = 27$ .
Multiply the index on $a$ by 3: $(a2)3=a6\left(a^2\right)^3 = a^6$ .
Remember $b$ means $b^1$ , so $(b1)3=b3\left(b^1\right)^3 = b^3$ .
Put everything together: $27a^6b^3$ .

Common Mistake

Forgetting to power the coefficient

In $(4x)2\left(4x\right)^2$ , the 4 is squared as well as the $x$ . So $(4x)2=16x2\left(4x\right)^2 = 16x^2$ , not $4x^2$ .

6. Zero powers

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

Key Idea

Zero index

a^0 = 1

This is true when $\neq 0$ .

Example

Using the zero power

Write down the value of $7^0$ .

The index is 0.
Any non-zero base to the power of 0 is 1.
So $7^0 = 1$ .

7. Negative powers and reciprocals

A reciprocal is what you multiply a number by to get 1. For example, the reciprocal of 5 is $15\frac{1}{5}$ .

Definition

Negative index

A negative index means “take the reciprocal”. For example:

a−n=1ana^{-n} = \frac{1}{a^n}

Example

Evaluating a negative power

Write down the value of $2^{-4}$ .

A negative index sends the power to the denominator as a reciprocal.

A negative index means write the power as a reciprocal.
So $2−4=1242^{-4} = \frac{1}{2^4}$ .
Work out $2^4 = 16$ .
The value is $116\frac{1}{16}$ .

Example

Finding a reciprocal

Write down the reciprocal of $14\frac{1}{4}$ .

The reciprocal flips the fraction over.
Flipping $14\frac{1}{4}$ gives $41\frac{4}{1}$ .
So the reciprocal is 4.

8. Unknown indices

Sometimes you are given an equation and asked to find a missing index.

The main idea is to use the index laws first, then match the powers.

Example

Finding a missing index in a product

Given $y3×ya=y8y^3 \times y^a = y^8$ , find $a$ .

The bases are the same, so add the indices on the left: $y^{3+a}$ .
This means $y^{3+a} = y^8$ .
Match the indices: $3 + a = 8$ .
Solve to get $a = 5$ .

Example

Finding a missing index in brackets

Given $(x5)b=x20\left(x^5\right)^b = x^{20}$ , find $b$ .

A power outside brackets means multiply the indices.
So $(x5)b=x5b\left(x^5\right)^b = x^{5b}$ .
Match the indices: $5 b = 20$ .
Solve to get $b = 4$ .

Example

Missing index in a division

Given $x5xa=x7\frac{x^5}{x^a} = x^7$ , find $a$ .

Dividing powers with the same base means subtract the indices.
So the left side becomes $x^{5-a}$ .
Match the indices: $5 - a = 7$ .
Solve to get $a = - 2$ .

9. Rewriting as a power of the same base

Sometimes you need to rewrite a number using the same base first.

For example, 64 can be written as $4^3$ , because $4^3 = 64$ .

Example

Writing as a power of 4

Write $16 \times 4^6$ as a power of 4.

Rewrite 16 as a power of 4: $16 = 4^2$ .
So the expression becomes $42×464^2 \times 4^6$ .
Add the indices: 2 + 6 = 8.
The answer is $4^8$ .

Example

Powers of 10

Given $1000^3 = 10^x$ , find $x$ .

Rewrite 1000 as a power of 10: $1000 = 10^3$ .
So $10003=(103)31000^3 = \left(10^3\right)^3$ .
Multiply the indices: 3 × 3 = 9.
Therefore $x = 9$ .

Exam technique

In the exam

Check whether the bases are the same before using an index law.
For brackets, apply the outside power to every factor inside, including the coefficient.
If an answer has a negative power, rewrite it as a reciprocal if the question asks for a value.

Self review

Check yourself

What is the simplified form of $a4×a7a^4 \times a^7$ ?
How would you simplify $(2x3y)2\left(2x^3y\right)^2$ ?
What is the reciprocal form of $5^{-2}$ ?

Indices

What you'll learn

What an index means and how to read powers like $x^5$
How to multiply and divide powers with the same base
How to simplify brackets such as $4a^2b)^3$
How zero powers, negative powers and reciprocals work

1. What is an index?

An index tells you how many times a number or letter is multiplied by itself.

In $5^3$ , the 5 is the base and the 3 is the index. You read it as “5 to the power of 3”.

The base is the number being multiplied, and the index shows how many copies of it are used.

Definition

Base and index

In $a^n$ , the base is $a$ and the index is $n$ . It means multiply $a$ by itself $n$ times.

So:

53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125

Example

Working out powers

Work out the value of $42×334^2 \times 3^3$ .

Each power is expanded first, then the two results are multiplied.

Work out each power separately: $4^2 = 16$ and $3^3 = 27$ .
Multiply the results: 16 × 27 = 432.
So $42×33=4324^2 \times 3^3 = 432$ .

2. Multiplying powers with the same base

When the base is the same, you add the indices.

Key Idea

Multiplying same-base powers

am×an=am+na^m \times a^n = a^{m+n}

For example, $x4×x2x^4 \times x^2$ means four lots of $x$ multiplied by two more lots of $x$ , making six lots altogether.

Multiplying same-base powers combines all the repeated factors, so the indices add.

Example

Simplifying a product

Simplify $x6×x5x^6 \times x^5$ .

Check the bases are the same. Both powers have base $x$ .
Add the indices: 6 + 5 = 11.
Write the answer as one power: $x^{11}$ .

Common Mistake

Do not multiply the indices

For $x6×x5x^6 \times x^5$ , the answer is $x^{11}$ , not $x^{30}$ . Multiplying powers with the same base means add the indices.

3. Dividing powers with the same base

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

Key Idea

Dividing same-base powers

aman=am−n\frac{a^m}{a^n} = a^{m-n}

This works when $\neq 0$ .

Example

Simplifying a division

Simplify $w9w4\frac{w^9}{w^4}$ .

Dividing powers with the same base cancels matching factors from the numerator and denominator.

Check the bases are the same. Both powers have base $w$ .
Subtract the indices: 9 − 4 = 5.
Write the answer as one power: $w^5$ .

You may also see division written with the divide symbol, such as $d8÷d3d^8 \div d^3$ . It means the same as a fraction.

Example

Dividing using the divide sign

Simplify $d10÷d6d^{10} \div d^6$ .

The base is the same, so subtract the indices.
Work out 10 − 6 = 4.
The simplified answer is $d^4$ .

4. Coefficients and letters together

A coefficient is the number in front of a letter. For example, in $7p^3$ , the coefficient is 7.

When simplifying expressions with numbers and letters, deal with the numbers first, then deal with each letter using index laws.

Example

Multiplying numbers and letters

Simplify $4m3n2×6mn54m^3n^2 \times 6mn^5$ .

Separate the coefficients and each letter base, then add indices only for matching letters.

Multiply the coefficients: 4 × 6 = 24.
For the $m$ terms, remember that $m$ means $m^1$ . So $m3×m=m4m^3 \times m = m^4$ .
For the $n$ terms, add the indices: $n2×n5=n7n^2 \times n^5 = n^7$ .
Put everything together: $24m^4n^7$ .

Tip

Invisible power

A letter on its own has an invisible index of 1. So $m$ means $m^1$ , and $q$ means $q^1$ .

A single letter has an understood power of 1, which is useful when applying index laws.

Example

Dividing numbers and letters

Simplify $18p5q46p2q\frac{18p^5q^4}{6p^2q}$ .

Divide the coefficients: 18 ÷ 6 = 3.
Subtract the indices for $p$ : $p5÷p2=p3p^5 \div p^2 = p^3$ .
Subtract the indices for $q$ : $q4÷q=q3q^4 \div q = q^3$ .
Put everything together: $3p^3q^3$ .

5. Powers outside brackets

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

The outside power acts on every factor inside the brackets, including any coefficient.

Key Idea

Power of a power

(am)n=amn\left(a^m\right)^n = a^{mn}

So you multiply the indices.

Example

Power of a power

Simplify $(t4)3\left(t^4\right)^3$ .

The base is $t$ , and there is a power raised to another power.
Multiply the indices: 4 × 3 = 12.
The answer is $t^{12}$ .

If there is a number and more than one letter inside the bracket, raise every part to the outside power.

Example

Brackets with numbers and letters

Simplify $(3a2b)3\left(3a^2b\right)^3$ .

Cube the coefficient: $3^3 = 27$ .
Multiply the index on $a$ by 3: $(a2)3=a6\left(a^2\right)^3 = a^6$ .
Remember $b$ means $b^1$ , so $(b1)3=b3\left(b^1\right)^3 = b^3$ .
Put everything together: $27a^6b^3$ .

Common Mistake

Forgetting to power the coefficient

In $(4x)2\left(4x\right)^2$ , the 4 is squared as well as the $x$ . So $(4x)2=16x2\left(4x\right)^2 = 16x^2$ , not $4x^2$ .

6. Zero powers

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

Key Idea

Zero index

a^0 = 1

This is true when $\neq 0$ .

Example

Using the zero power

Write down the value of $7^0$ .

The index is 0.
Any non-zero base to the power of 0 is 1.
So $7^0 = 1$ .

7. Negative powers and reciprocals

A reciprocal is what you multiply a number by to get 1. For example, the reciprocal of 5 is $15\frac{1}{5}$ .

Definition

Negative index

A negative index means “take the reciprocal”. For example:

a−n=1ana^{-n} = \frac{1}{a^n}

Example

Evaluating a negative power

Write down the value of $2^{-4}$ .

A negative index sends the power to the denominator as a reciprocal.

A negative index means write the power as a reciprocal.
So $2−4=1242^{-4} = \frac{1}{2^4}$ .
Work out $2^4 = 16$ .
The value is $116\frac{1}{16}$ .

Example

Finding a reciprocal

Write down the reciprocal of $14\frac{1}{4}$ .

The reciprocal flips the fraction over.
Flipping $14\frac{1}{4}$ gives $41\frac{4}{1}$ .
So the reciprocal is 4.

8. Unknown indices

Sometimes you are given an equation and asked to find a missing index.

The main idea is to use the index laws first, then match the powers.

Example

Finding a missing index in a product

Given $y3×ya=y8y^3 \times y^a = y^8$ , find $a$ .

The bases are the same, so add the indices on the left: $y^{3+a}$ .
This means $y^{3+a} = y^8$ .
Match the indices: $3 + a = 8$ .
Solve to get $a = 5$ .

Example

Finding a missing index in brackets

Given $(x5)b=x20\left(x^5\right)^b = x^{20}$ , find $b$ .

A power outside brackets means multiply the indices.
So $(x5)b=x5b\left(x^5\right)^b = x^{5b}$ .
Match the indices: $5 b = 20$ .
Solve to get $b = 4$ .

Example

Missing index in a division

Given $x5xa=x7\frac{x^5}{x^a} = x^7$ , find $a$ .

Dividing powers with the same base means subtract the indices.
So the left side becomes $x^{5-a}$ .
Match the indices: $5 - a = 7$ .
Solve to get $a = - 2$ .

9. Rewriting as a power of the same base

Sometimes you need to rewrite a number using the same base first.

For example, 64 can be written as $4^3$ , because $4^3 = 64$ .

Example

Writing as a power of 4

Write $16 \times 4^6$ as a power of 4.

Rewrite 16 as a power of 4: $16 = 4^2$ .
So the expression becomes $42×464^2 \times 4^6$ .
Add the indices: 2 + 6 = 8.
The answer is $4^8$ .

Example

Powers of 10

Given $1000^3 = 10^x$ , find $x$ .

Rewrite 1000 as a power of 10: $1000 = 10^3$ .
So $10003=(103)31000^3 = \left(10^3\right)^3$ .
Multiply the indices: 3 × 3 = 9.
Therefore $x = 9$ .

Exam technique

In the exam

Check whether the bases are the same before using an index law.
For brackets, apply the outside power to every factor inside, including the coefficient.
If an answer has a negative power, rewrite it as a reciprocal if the question asks for a value.

Self review

Check yourself

What is the simplified form of $a4×a7a^4 \times a^7$ ?
How would you simplify $(2x3y)2\left(2x^3y\right)^2$ ?
What is the reciprocal form of $5^{-2}$ ?

Indices

What you'll learn

1. What is an index?

2. Multiplying powers with the same base

3. Dividing powers with the same base

4. Coefficients and letters together

5. Powers outside brackets

6. Zero powers

7. Negative powers and reciprocals

8. Unknown indices

9. Rewriting as a power of the same base

Recap questions

You've reached the end

Indices

What you'll learn

1. What is an index?

2. Multiplying powers with the same base

3. Dividing powers with the same base

4. Coefficients and letters together

5. Powers outside brackets

6. Zero powers

7. Negative powers and reciprocals

8. Unknown indices

9. Rewriting as a power of the same base

Recap questions

You've reached the end

Indices

What you'll learn

1. What is an index?

2. Multiplying powers with the same base

3. Dividing powers with the same base

4. Coefficients and letters together

5. Powers outside brackets

6. Zero powers

7. Negative powers and reciprocals

8. Unknown indices

9. Rewriting as a power of the same base

Recap questions

You've reached the end

Indices Revision Guide

Indices

What you'll learn

1. What is an index?

2. Multiplying powers with the same base

3. Dividing powers with the same base

4. Coefficients and letters together

5. Powers outside brackets

6. Zero powers

7. Negative powers and reciprocals

8. Unknown indices

9. Rewriting as a power of the same base

Recap questions

You've reached the end

Indices Revision Guide