Powers and Roots Worksheets | Questions and Revision | MME

Powers and Roots Worksheets, Questions and Revision

Level 4-5

Powers and Roots

Powers are a shorthand way of expressing repeated multiplication. Roots are ways of reversing this. There are a total of 10 indices rules. This page will give you the 7 easy rules to remember; there are 3 further more complex rules which can be found in the laws of indices page. 

Make sure you are happy with the following topics before continuing.

Level 4-5
KS3 Level 4-5

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Indices Rule 1: The Multiplication Law

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

a^\textcolor{red}{b} \times a^\textcolor{blue}{c} = a^{\textcolor{red}{b} + \textcolor{blue}{c}}

This multiplication law applies to all terms with powers (positive or negative): e.g.

x^{\textcolor{red}{-m}}\times x^\textcolor{blue}{n}=x^{({\textcolor{red}{-m})}\textcolor{blue}{+n}}=x^{\textcolor{blue}{n}\textcolor{red}{-m}}

This works for fractional powers too. Remember when adding fractions, they must  share a common denominator. 

x^{\textcolor{red}{\frac{1}{3}}} \times x^{\textcolor{blue}{\frac{1}{6}}}=x^{\textcolor{red}{\frac{2}{6}}+\textcolor{blue}{\frac{1}{6}}} = x^{\textcolor{black}{\frac{3}{6}}} = x^{\textcolor{black}{\frac{1}{2}}}

KS3 Level 4-5

Indices Rule 2: The Division Law

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

a^\textcolor{red}{b} \div a^\textcolor{blue}{c} = a^{\textcolor{red}{b} - \textcolor{blue}{c}}

The division law applies to all numbers, negative numbers and fractional powers,

x^\textcolor{red}{6}\div x^\textcolor{blue}{2}=\dfrac{x^\textcolor{red}{6}}{x^\textcolor{blue}{2}}=x^{\textcolor{red}{6} - \textcolor{blue}{2}} = x^{4}

KS3 Level 4-5

Indices Rule 3: Multiple Powers Law

The multiple powers law is when you raise one power to another, i.e. the power of a power. When this happens the powers are multiplied:


A basic example shows how the multiple powers law works with numbers:


KS3 Level 4-5
Level 4-5

Indices Rule 4: Power 0 Law

Anything to the power 0 = 1

a^\textcolor{blue}{0} = \textcolor{red}{1}

The power 0 law applies to everything: 100^\textcolor{blue}{0}=\textcolor{red}{1}, \quad x^\textcolor{blue}{0}=\textcolor{red}{1} \quad  \pi^\textcolor{blue}{0}=\textcolor{red}{1}

Level 4-5

Indices Rule 5: Power 1 Law

Anything to the power 1 is just itself. 

\textcolor{red}{a}^\textcolor{blue}{1} = \textcolor{red}{a}

The power 1 law applies to everything: \textcolor{red}{100}^\textcolor{blue}{1}=\textcolor{red}{100}, \quad \textcolor{red}{x}^\textcolor{blue}{1}=\textcolor{red}{x}, \quad \textcolor{red}{\pi}^\textcolor{blue}{1}=\textcolor{red}{\pi}

Level 4-5

Indices Rule 6: The 1 Law

1 to the power anything = 1 e.g.

\textcolor{red}{1}^\textcolor{blue}{x} =\textcolor{red}{1}

This works for any power: \textcolor{red}{1}^\textcolor{blue}{100} =\textcolor{red}{1}, \quad \textcolor{red}{1}^\textcolor{blue}{-5} =\textcolor{red}{1}

Level 4-5
Level 4-5

Indices Rule 7: The Fraction Law 

The power of a fraction applies to both the top and bottom of the fraction. 

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

This also applies to mixed factions 

\bigg(2\dfrac{\textcolor{red}{3}}{\textcolor{blue}{4}}\bigg)^\textcolor{limegreen}{5} = \bigg(\dfrac{\textcolor{red}{11}}{\textcolor{blue}{4}}\bigg)^\textcolor{limegreen}{5} = \bigg(\dfrac{\textcolor{red}{11}^\textcolor{limegreen}{5}}{\textcolor{blue}{4}^\textcolor{limegreen}{5}}\bigg)

Level 4-5


The opposite to taking a power of some number is to take a root. Let’s consider square roots – these do the opposite of squaring. e.g.

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

\sqrt[\textcolor{red}{2}]{\textcolor{limegreen}{16}} = \textcolor{blue}{4}

We also have cube roots, 4th roots, 5th roots, etc, for when the powers are higher. e.g.

\textcolor{blue}{2}^\textcolor{red}{3} = \textcolor{limegreen}{8}

\sqrt[\textcolor{red}{3}]{\textcolor{limegreen}{8}} = \textcolor{blue}{2}

These roots use the same symbol, just with a different number in the top left to show the power, e.g. \sqrt[3]{8}=2

A 4th root would be shown by \sqrt[4]{}, and so on.

KS3 Level 4-5
Level 4-5

Example 1: Multiplication

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

[2 marks]

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, putting like terms together. 

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

Finally using rule 1 we can multiply the following,


p^2\times p=p^3

q^3\times q^4=q^7

This gives the final answer to be,


KS3 Level 4-5

Example 2: Multiplication and Division

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

[2 marks]

First we must multiply out the top of the fraction,


So, the calculation becomes


Next calculating the division we get, 


This gives the final answer to be,


Level 4-5

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Example Questions

we know that: 

a^a \times a^c = a^{b + c}


a^2 \times a^3 = a^{2+3}

a^2 \times a^3 = a^5

It is helpful to be able to recognise the first 15 square numbers. 


In this case, we can recognise, 


12^2=144 and 14^2=196


Hence the calculation is simply, 




We can rewrite the first term of the expression as, 




The multiplication law tells us that, 




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




Hence, the expression now looks like, 




Using the division law we find, 



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




Thus the expression now is, 




This can be simplified to, 




Hence we are left with a simple calculation of, 



We know that,

20^1 = 20


100^0 = 1

So we can calculate 

20 + 1 = 21

Worksheets and Exam Questions


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Drill Questions


Powers And Square roots - Drill Questions