Indices Rules Worksheets, Questions and Revision (2024)

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.

• Collecting like terms
• Powers and roots

Level 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

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 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

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}

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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 4-5GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

Level 4-5GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

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

4^{-3}=\frac{1}{64}.

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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 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

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,

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

Next, using Rule 3, we can simplify,

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

So the whole expression can be written as

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

Finally using Rule 1we simplify the expression further.

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

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

2^3 = 8

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Rules of Indices Example Questions