Indices Rules Worksheets | Questions and Revision | MME

Indices Rules Worksheets, Questions and Revision

Level 6-7

Indices Rules

Indices Rules builds on the 7 rules from Powers and Roots. We will cover 3 more complicated rules here. Make sure you are confident with the following topics before moving onto laws and indices

Level 6-7
Level 6-7

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Indices Rule 8: Fractional Powers

The fractional indices laws apply when the power is a fraction. 

\textcolor{red}{a}^{\large{\frac{\textcolor{blue}{b}}{\textcolor{limegreen}{c}}}} = \sqrt[\textcolor{limegreen}{c}]{\textcolor{red}{a}^\textcolor{blue}{b}}

This is commonly use to show square and cube roots

\textcolor{red}{x}^{\large{\frac{\textcolor{limegreen}{1}}{\textcolor{blue}{2}}}}= \sqrt[\textcolor{blue}{2}]{\textcolor{red}{x}^\textcolor{limegreen}{1}} =\sqrt[\textcolor{blue}{2}]{\textcolor{red}{x}}

\textcolor{red}{x}^{\large{\frac{\textcolor{limegreen}{1}}{\textcolor{blue}{3}}}}= \sqrt[\textcolor{blue}{3}]{\textcolor{red}{x}^\textcolor{limegreen}{1}} =\sqrt[\textcolor{blue}{3}]{\textcolor{red}{x}}

Note: it doesn’t matter which order you carry out the square root and multiplication operations. In other words, the rule can also be written as

\textcolor{red}{a}^{\large{\frac{\textcolor{blue}{b}}{\textcolor{limegreen}{c}}}} = (\sqrt[\textcolor{limegreen}{c}]{\textcolor{red}{a}})^\textcolor{blue}{b}

You should try to carry out the operations in the order that makes the calculation as simple as possible.

Level 6-7

Indices Rule 9: Multi-step Fractional Powers

You may also be asked to simplify expressions where the numerator is not \bf{1}

\textcolor{red}{64}^{\large{\frac{\textcolor{limegreen}{2}}{\textcolor{blue}{3}}}}= \sqrt[\textcolor{blue}{3}]{\textcolor{red}{64}^\textcolor{limegreen}{2}}

\sqrt[\textcolor{blue}{3}]{\textcolor{red}{64}} = \textcolor{red}{4}

\textcolor{red}{4}^\textcolor{limegreen}{2} = \textcolor{red}{16}

Level 6-7

Indices Rule 10: Negative Powers

Negative powers flip the fraction and put 1 over the number 

In general, the result of a negative power is “\bf{1} over that number to the positive power”, i.e.

\textcolor{red}{a}^{-\textcolor{limegreen}{b}} = \dfrac{1}{\textcolor{red}{a}^\textcolor{limegreen}{b}}

for any value of a or b. When the power is \textcolor{blue}{-1}, this takes the form, 

\textcolor{red}{a}^{\textcolor{blue}{-1}}=\dfrac{1}{\textcolor{red}{a}} or \textcolor{red}{10}^{\textcolor{blue}{-1}} = \dfrac{1}{\textcolor{red}{10}}

When the number is a fraction, the negative power flips the fraction

\bigg(\dfrac{\textcolor{blue}{a}}{\textcolor{limegreen}{b}}\bigg)^{-\textcolor{red}{x}} = \bigg(\dfrac{\textcolor{limegreen}{b}}{\textcolor{blue}{a}}\bigg)^\textcolor{red}{x} 

Level 6-7
Level 6-7

Example 1: Negative Powers

Simplify the following,  4^{-3}.

[2 marks]

We now know that 4^{-3} is equal to \dfrac{1}{4^3}. We also know that

4^3=4\times 4\times 4=16\times 4=64.

So, we get that


Level 6-7

Example 2: Fractional Powers and Roots

Simplify the following,  9^{\frac{3}{2}}.

[2 marks]

So, we know that 9^{\frac{3}{2}} is equal to \sqrt[2]{9^3} or (\sqrt[2]{9})^3

So, to work out (\sqrt[2]{9})^3, we first have to square root 9, which is easy enough – the square root of 9 is 3. So, (\sqrt[2]{9})^3 becomes 3^3, which is

3^3=3\times 3\times 3 = 27

Level 6-7

Example 3: Multiplication and Powers

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

[3 marks]

The first part of the expression is a power of 2, whilst the second part is a power of 8.

we know that

8 = 2^3

This means we can rewrite the following,


Next, using Rule 3, we can simplify,


So the whole expression can be written as


Finally using Rule 1 we simplify the expression further. 


Thus, we have written the expression as a power of 2. Evaluating this final answer gives

2^3 = 8

Level 6-7

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Example Questions

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




Using the power law, we get




Therefore, the whole expression becomes




Applying the multiplication law, this simplifies to




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

Firstly, as 3^2=9, the inverse operation gives, \sqrt{9}=3


So, that leaves 6^{-2}, this becomes the following fraction,




We know that 6^2=6\times 6=36, so 




Multiplying our two answers together, we get


\sqrt{9}\times 6^{-2}=3\times\dfrac{1}{36}=\dfrac{3}{36}=\dfrac{1}{12}

This expression can be rewritten as, 


\sqrt4 \times (\sqrt4)^3 


Given we know that \sqrt4=2 , this becomes, 







Notice that in this example we chose to perform the \sqrt{4} operation before cubing the answer. We could alternatively write the expression as \sqrt{4^3}, but in this case the first option is easier.

As it is a negative power we can rewrite this as, 




Now, we can work out the denominator, which we will write as, 




We know that \sqrt[3]{8}=2. So this simplifies to,




Counting up in powers of 2: 4, 8, 16, 32 – we see that 32 is the 5th power of 2, so




Therefore, the answer is,



Worksheets and Exam Questions


(NEW) Rules of Indices Exam Style Questions - MME

Level 6-7 New Official MME

Drill Questions


Indices Rules - Drill Questions


Fractional And Negative Indices - Drill Questions


Rules of Indices - Drill Questions