## What you need to know

### Indices Rules

There are three rules of indices (or laws of indices) which you have to know and be able to apply to problems involving both numbers and algebra. For any numbers, x, m, and n, those three rules are

- The multiplication law – when you multiply terms, you add the powers:

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

- The division law – when you divide terms, you subtract the powers:

x^m\div x^n=\dfrac{x^m}{x^n}=x^{m-n}

- The power law – when you take a power of a term already with a power, the powers are multiplied:

\left(x^m\right)^n=x^{mn}

Be confident with powers and roots will help you to master the laws of indices.

### Indices Rule 1: The multiplication Law

The multiplication law states when you multiply terms, you add the powers as shown

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

This application of the multiplication law is used all the time when you expand brackets as well as other standard algebra practises. With practice, you should be able to perform this process quite quickly and apply it other variations of the question for example when the powers are negative.

x^{-m}\times x^n=x^{n-m}

Other applications of the multiplication law of indices include fractional powers

x^{\frac{1}{3}}\times x^{\frac{1}{6}}=x^{\frac{1}{2}}

Here we made the fractional powers have the same denominator in order to be able to add them

\frac{2}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}

You will need to learn to apply all of the indices laws in different ways to be able to access the higher-level questions.

### Indices Rule 2: The Division Law

The division law is when you divide terms and in doing so, you subtract the powers:

x^m\div x^n=\dfrac{x^m}{x^n}=x^{m-n}

Like the multiplication law this applies to algebra, numbers, negative numbers and fractional powers. The most difficult indices questions use a combination of the different rules so you need to be confident in order to apply them in the correct order.

A basic example with numbers shows how the division law works.

x^6\div x^2=\dfrac{x^6}{x^2}=x^{4}

### Indices Rule 3: The Power Law

The power law is when you have a power of a term already with a power, so a power of a power. When this happens the powers are multiplied:

\left(x^m\right)^n=x^{mn}

As with the other indices laws you will need to learn to apply these to different types of indices questions.

A basic example shows how the power law works with numbers

\left(x^3\right)^2=x^{6}

### Example 1: Multiplication

Write 5p^2q^3\times3pq^4 in its simplest form.

To simplify this expression, we must recognise that it can be broken up into parts, i.e. we can write

5p^2q^3\times3pq^4=5\times p^2\times q^3\times3\times p\times q^4

Then, we can rearrange the terms of this multiplication to make it

5\times 3\times p^2\times p\times q^3\times q^4

Firstly, 5\times3=15. Then, the rest of the terms can be simplified using the multiplication law. Recalling that p=p^1, we get that

p^2\times p=p^3\,\text{ and }\,q^3\times q^4=q^7

Therefore, the expression simplifies to

15p^3q^7

### Example 2: Multiplication and Division

Work out the value of \dfrac{3^4\times3^7}{3^8}. (Non calculator)

Applying the multiplication law, we get that

3^4\times3^7=3^{4+7}=3^{11}

So, the calculation becomes

\dfrac{3^{11}}{3^8}

This is a division, so applying the division law, we get

\dfrac{3^{11}}{3^8}=3^{11-8}=3^3

The final answer is

3^3=27.

**Note:** problems like this can only be simplified if the base is the same, i.e. we can do 3^4\times3^7=3^{11}, but there is no good way to simplify 3^4\times2^5 since their bases are different.

### Example 3: Multiplication and Powers

Write 2^{15}\times 8^{-4} as a power of 2, and hence evaluate the expression. (Non calculator)

The first part of the expression is a power of 2, whilst the second part is a power of 8. The key observation here is realising that 8 is actually a power of 2, specifically it is 2^3. This means we can write this as

8^{-4}=\left(2^3\right)^{-4}

Now, this is in the appropriate form for us to apply the power law. Doing so, we get

\left(2^3\right)^{-4}=2^{3\times(-4)}=2^{-12}

So the whole expression can be written as

2^{15}\times2^{-12},

we can use the multiplication law to get

2^{15}\times2^{-12}=2^{15+(-12)}=2^3

Thus, we have written the expression as a power of 2. Evaluating the expression: 2^3, we get 2\times2\times2=8.

**Note: **if you’re taking the higher paper, there’s no reason why these types of questions can’t include fractional powers. That said, the ideas are exactly the same, you just have to be more careful with your calculations, since fractions make everything a bit more effort.

### Example Questions

1) Work out (3^2)^3\div3^4

(Non-calculator)

We can rewrite the first term of the expression as,

(3^2)^3=3^2\times3^2\times3^2

The multiplication law tells us that,

3^2\times3^2\times3^2=3^{2+2+2}=3^6

This is the same result as the power-law gives,

(3^2)^3=3^{2\times3}=3^6

Hence, the expression now looks like,

3^6\div3^4

Using the division law we find,

3^6\div3^4=3^{6-4}=3^2=9

2) Work out \dfrac{7^5\times7^3}{7^6}

(Non-calculator)

First considering the numerator, the laws of indices tell us,

7^5\times7^3=7^{5+3}=7^8

Thus the expression now is,

\dfrac{7^8}{7^6}

This can be simplified to,

\dfrac{7^8}{7^6}=7^{8-6}=7^2

Hence we are left with a simple calculation of,

7^2=7\times7=49

3) Work out the value of \dfrac{5^9}{5^{14} \div 5^7}

(Non-calculator)

Let’s consider the denominator, we can apply the division law so that,

5^{14}\div5^7=5^{14-7}=5^7

Therefore, the expression becomes,

\dfrac{5^9}{5^7}

So, we can apply the division law again. Doing so, we get,

\dfrac{5^9}{5^7}=5^{9-7}=5^2

Thus, the value of the expression is

5^2=25

4) Simplify the expression,

\dfrac{abc^2\times a^{3}c}{ab^2\times\left(c^2\right)^3}

First, we will simplify the numerator. Breaking up the components, we can write it as

a\times a^3\times b\times c^2\times c

Applying the multiplication law, this becomes

a^{4}bc^{3}

Next, if we apply the power law to the denominator then we get that (c^2)^3=c^{2\times3}= c^6

Thus, the expression is,

\dfrac{a^{4}bc^{3}}{ab^{2}c^{6}}

Cancelling powers between the numerator and the denominator we are left with,

\dfrac{a^{\cancel{4}3}\cancel{b}c^{\cancel3}}{\cancel{a}b^{\cancel2}c^{\cancel{6}3}}=\dfrac{a^3}{bc^3}

5) Write 9^5\times3^{-5} as a power of 3

(Non-calculator)

So, we can’t use any laws straight away since the terms don’t have the same base. However, if we recognise that 9=3^2, then we can write the first term as

\left(3^2\right)^5

Using the power law, we get

\left(3^2\right)^5=3^{2\times5}=3^{10}

Therefore, the whole expression becomes

3^{10}\times3^{-5}

Applying the multiplication law, this simplifies to

3^{10+(-5)}=3^5

Thus, we have written the expression as a power of 3.

### Worksheets and Exam Questions

#### (NEW) Rules of Indices Exam Style Questions - MME

Level 6-7#### Indices Rules - Drill Questions

Level 4-5#### Fractional And Negative Indices - Drill Questions

Level 6-7#### Rules of Indices - Drill Questions

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