Powers and Roots Worksheets | Questions and Revision | MME

Powers and Roots Worksheets, Questions and Revision

Level 4-5
Maths Exam Worksheets

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

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}}}

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}

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:

\left(a^\textcolor{red}{b}\right)^\textcolor{limegreen}{c}=a^{\textcolor{red}{b}\textcolor{limegreen}{c}}

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

\left(x^\textcolor{red}{3}\right)^\textcolor{limegreen}{2}=x^{\textcolor{red}{3}\times\textcolor{limegreen}{2}}=x^{6}

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

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

Roots

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.

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,

5\times3=15

p^2\times p=p^3

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

This gives the final answer to be,

15p^3q^7

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,

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

So, the calculation becomes

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

Next calculating the division we get, 

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

This gives the final answer to be,

3^3=27.

Level 4-5
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Example Questions

we know that: 

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

so,

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, 

 

\sqrt{144}+\sqrt{196}=12+14=26

 

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

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

We know that,

20^1 = 20

and 

100^0 = 1

So we can calculate 

20 + 1 = 21

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