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Here we will learn everything you need to know about the laws of indices for GCSE & iGCSE maths (Edexcel, AQA and OCR). You’ll learn what the laws of indices are and how we can use them. You’ll learn how to multiply indices, divide indices, use brackets and indices, how to raise values to the power of 0 and to the power of 1, as well as fractional and negative indices.

Look out for the index laws worksheet and exam questions at the end.

An **index** is a small number that tells us how many times a term has been multiplied by itself.

The plural of **index** is **indices.**

Below is an example of a term written in index form:

\[4^{3}\]

**4** is the base and **3** is the index.

We can read this as ‘4 to the power 3’

Another way of expressing 4^{3} is

\[4 \times 4 \times 4=64\]

Indices can be positive or negative numbers.

Index Form

\[ 2^{3} \]

\[2^{2}\]

\[2^{1}\]

\[2^{0}\]

\[2^{-1}\]

\[2^{-2}\]

\[2^{-3}\]

Expanded Form

\[2 \times2 \times 2\]

\[2\times 2\]

\[2\]

\[1\]

\[\frac{1}{2}\]

\[\frac{1}{2^{2}}=\frac{1}{2} \times \frac{1}{2}\]

\[\frac{1}{2^{3}}=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\]

Value

\[8\]

\[4\]

\[2\]

\[1\]

\[\frac{1}{2}=0.5\]

\[\frac{1}{4}=0.25\]

\[\frac{1}{8}=0.125\]

Moving up the rows we get 2 times larger per row.

Moving down the rows we get 2 times smaller per row.

In order to calculate with indices we need to be able to use the laws of indices in a variety of different ways. Let’s look at the different ways we can calculate with indices.

Laws of indices provide us with rules for simplifying calculations or expressions involving powers of the **same base**. This means that the larger number or letter must be the same.

For example,

\[2^{5} \times 2^{3}=2^{8}\]

\[x^{10} \div x^{4}=10^{6}\]

We cannot use laws of indices to evaluate calculations when the bases are different

For example,

\[2^{5} \times 3^{3}\]

\[x^{10} \div y^{4}\]

We can however evaluate these calculations. Check out our other pages to find out how.

There are several laws of indices (sometimes called indices rules), including multiplying, dividing, power of 0, brackets, negative and fractional powers.

For examples and practice questions on each of the rules of indices, as well as how to evaluate calculations with indices with different bases, follow the links below.

**See also**: Index notation

When multiplying indices with the same base, add the powers.

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

**Step-by-step guide: **Multiplying indices

When dividing indices with the same base, subtract the powers.

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

**Step-by-step guide: **Dividing indices

When there is a power outside the bracket multiply the powers.

\[(a^{m})^{n}=a^{m \times n}\]

**Step-by-step guide:** Brackets with indices

Any non-zero value raised to the power of 0 is equal to 1.

\[a^{0}=1\]

**Step-by-step guide:** Power of 0

When the index is negative, put it over 1 and flip (write its reciprocal) to make it positive.

\[a^{-m}=\frac{1}{a^{m}}\]

**Step-by-step guide: **Negative indices

**When the index is a fraction, the denominator is the root of the number or letter, then raise the answer to the power of the numerator.**

x^{\frac{a}{b}}=(\sqrt[b]{x})^{a}

**Step-by-step guide: **Fractional indices

Get your free laws of indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free laws of indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEWe can use the laws of indices to simplify expressions.

**When multiplying indices with the same base, add the powers.**

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

1 **Add the powers**.

2 **Multiply any coefficients**.

…………………………………………………………………………………………………………………………………………………….

Simplify

\[4 a^{3} \times 7 a^{2}\]

**This can be written as**

\[4 a^{3} \times 7 a^{2}=4 \times a \times a \times a \times 7 \times a \times a\]

1 **Add together the indices 3 and the 2.**

\[3 + 2 = 5\]

2 **Multiply 4 and 7 together.**

\[4 \times 7 =28\]

So,

\[4 a^{3} \times 7 a^{2}=28 a^{5}\]

**When dividing indices with the same base, subtract the powers.**

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

**Subtract the indices**.**Divide any coefficients of the base number or letter**.

…………………………………………………………………………………………………………………….………………………………

Simplify

\[21 a^{9} \div 7 a^{2}\]

**Subtract the indices 9 and 2.**

\[9 – 2 = 7\]

**Divide the coefficients 21 and 7.**

\[21 ÷ 7 = 3\]

So,

\[21 a^{9} \div 7 a^{2}=3a^{7}\]

**When there is a power outside the bracket multiply the powers.**

\[(a^{m})^{n}=a^{m \times n}\]

**Multiply the indices**.**Raise all coefficients inside the bracket to the power outside the bracket**.

Why does this work?

Let’s look at

\[(a^{3})^{4}\]

This algebraic expression has been raised to the power of 4, which means:

\[a^{3} \times a^{3} \times a^{3} \times a^{3}\]

We know when we multiply indices with the same base, we must add the powers.

\[a^{3} \times a^{3} \times a^{3} \times a^{3}=a^{12}\]

Is there a quicker way to work this out?

Simplifying ^{3} × a^{3} × a^{3} × a^{3}^{3×4} we can work out the simplified answer is ^{12}

\[(a^{3})^{4}=a^{3 \times 4}=a^{12}\]

When we have brackets and indices we can multiply the powers.

Simplify and leave your answer in index form.

\[(a^{3})^{4}\]

**Multiply the powers 3 and the 4.**

\[3\times 4=12\]

So,

\[\left(a^{3}\right)^{4}=a^{12}\]

Simplify and leave your answer in index form.

\[(10 a^{3})^{2}\]

**Multiply the indices 3 and the 2.**

\[3\times 2=6\]

**Raise 10 to the power of 2.**

\[10^{2}=100\]

So,

\[\left(10 a^{3}\right)^{2}=100 a^{6}\]

Simplify and leave your answer in index form.

\[(-4a^{-3})^{2}\]

**Multiply the indices -3 and the 2.**

\[-3\times 2=-6\]

**Raise -4 to the power of 2.**

\[(-4)^{2}=16\]

So,

\[\left(-4 a^{-3}\right)^{2}=16 a^{-6}\]

**Any non-zero value raised to the power of 0 is equal to 1**.

Simplify

\[12a^{0}\]

\[= 12 (1) = 12\]

We can see that

Multiplying anything by 1 leaves it unchanged, this is called the multiplicative identity.

So,

\[12 a^{0}=12\]

**When the index is negative, put it over 1 and flip to make it positive.**

\[a^{-m}=\frac{1}{a^{m}}\]

**Put the term over 1**.**Flip the fraction to make the index positive**.**Simplify if necessary**.

Simplify

\[(2a)^{-3}\]

Notice how the index affects the entire bracket

**Put the term over 1.**

\[\frac{(2a)^{-3}}{1}\]

**Flip and change the power -3 to +3.**

\[\frac{1}{(2a)^{3}}\]

**Simplify the denominator.**

\[=\frac{1}{8a^{3}}\]

So,

\[(2 a)^{-3}=\frac{1}{8 a^{3}}\]

**When the index is a fraction, the denominator is the root of the number or letter, then raise the answer to the power of the numerator.**

\[x^{\frac{a}{b}}=(\sqrt[b]{x})^{a}\]

**Use the denominator to find the root of the term****Raise the answer to the power of the numerator**

Evaluate

\[27^{\frac{2}{3}}\]

**Use the denominator to find the cube root of the number or letter.**

\[\sqrt[3]{27}=3\]

**Raise the answer to the power of the numerator.**

\[3^{2}=9\]

**So, **

\[27^{\frac{2}{3}}=9\]

**The multiplication and division laws can only be used for terms with the same base**

You cannot simplify

\[a^{4} \times b^{3}\]

as the bases are different

**When a power is outside the bracket you must raise all terms inside the bracket by that power**

E.g.

When we simplify

\[(4a)^{2}\]

We must remember to square both the 4 and the a. It is common to forget to square the 4.

\[(4 a)^{2}=4 a \times 4 a=4^{2} a^{2}=16 a^{2}\]

The brackets mean the entire term is squared.

**Confusing integer and fractional powers**

Raising a term to the power of 2 means we square it

E.g.

\[a^{2}=a \times a\]

Raising a term to the power of ½ means we find the square root of it

E.g.

\[a^{\frac{1}{2}}=\pm \sqrt{a}\]

Raising a term to the power of 3 means we cube it

E.g.

\[a^{3}=a \times a \times a\]

Raising a term to the power of ⅓ means we find the cube root of it

E.g.

\[a^{\frac{1}{3}}=\sqrt[3]{a}\]

**Confusing positive and negative powers**

We can have decimal, fractional, negative or positive integers.

**Indices, powers or exponents**

Indices can also be called powers or exponents.

1. Simplify

4 a^{6} \times 5 a^{-2}

20 a^{8}

20 a^{-12}

20 a^{4}

20 a^{12}

We need to multiply the coefficients

4\times5=20

Since the base number is the same in each term, we can add the indices

6+(-2)=4

This means

4 a^{6} \times 5 a^{-2}=20 a^{4}

2. Simplify

-3 a^{4 b} \times 6 a^{7 b}

18 a^{11 b}

18 a^{28 b}

-18 a^{28 b}

-18 a^{11 b}

We need to multiply the coefficients

-3\times6=-18

Since the base number is the same in each term, we can add the indices

4b+7b=11b

This means

-3 a^{4 b} \times 6 a^{7 b}=-18 a^{11 b}

3. Simplify

20 b^{10} \div 4 b^{5}

5 b^{2}

5 b^{5}

16 b^{5}

20 b^{5}

We need to multiply the coefficients

20\div4=5

Since the base number is the same in each term, we can subtract the indices

10-5=5

This means

20 b^{10} \div 4 b^{5}=5 b^{5}

4. Simplify

42a^{-3} \div 6 a^{-8}

7 a^{-5}

7 a^{11}

7 a^{5}

7 a^{-11}

We need to divide the coefficients

42\div6=7

Since the base number is the same in each term, we can subtract the indices

-3- \,-8=-3+8=5

This means

42a^{-3} \div 6 a^{-8}=7 a^{5}

5. Simplify

(32 x)^{0} \times 6^{2}

36x^{2}

192x

36

192x^{2}

Anything raised to the power zero is 1 , so

(32 x)^{0}=1

This means we have

1\times6^{2}=1\times36=36

6. Simplify

(5 x)^{-3}

125x^{-3}

125x^{3}

\frac{1}{125x^{-3}}

\frac{1}{125x^{3}}

The negative index number indicates we need the reciprocal of

(5x)^{3}

Since

5x^{3}=5x\times5x\times5x=125x^{3}

We have

(5 x)^{-3}=\frac{1}{125 x^{3}}

7. Simplify

6 b^{-4}

2b

\frac{3}{2b}

\frac{6}{b^{4}}

\frac{1296}{b^{4}}

The negative index number indicates we need the reciprocal of

b^{4}

The index number does not affect the 6 , hence

6 b^{-4}=\frac{6}{b^{4}}

8. Evaluate

64^{\frac{2}{3}}

8

16

512

\frac{128}{3}

The index number is telling us to square the cube root, so

64^{\frac{2}{3}}=(\sqrt[3]{64})^{2}=4^{2}=16

9. Evaluate

16^{-\frac{1}{2}}

8

\frac{1}{4}

4

-8

The index number is telling us to find the reciprocal of the square root, so

\begin{aligned} &16^{-\frac{1}{2}}\\ &=\frac{1}{16^{\frac{1}{2}}}\\ &=\frac{1}{\sqrt{16}}\\ &=\frac{1}{4} \end{aligned}

1. Simplify

\frac{p^{7}}{p^{3}}

**(1 mark)**

Show answer

p^{4}

**(1)**

2. Simplify

6 h^{3} m^{6} \times 4 h^{4} m^{5}

**(2 marks)**

Show answer

h^{7} \text { or } m^{11} seen (evidence of adding powers)

**(1)**

24 h^{7} m^{11}

**(1)**

3. Simplify

\frac{12 x^{5} y^{7}}{3 x^{2} y}

**(2 marks)**

Show answer

x^{3} \text { or } y^{6} seen (evidence of subtracting powers)

**(1)**

4 x^{3} y^{6}

**(1)**

You have now learned how to:

- Simplify expressions involving the laws of indices
- Calculate with roots, and with integer and fractional indices

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