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Here we will learn how to simplify and evaluate with fractional indices for GCSE maths (Edexcel, AQA and OCR).

Look out for the laws of indices worksheets and exam questions at the end.

Fractional indices are powers of a term that are fractions. Both parts of the fractional power have a meaning.

\[x^{\frac{a}{b}}\]

The **denominator** of the fraction (b) is the **root** of the number or letter.

The **numerator** of the fraction (a) is the power to raise the answer to.

\[(\sqrt[b]{x})^{a}\]

Get your free fractional indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free fractional indices worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEFractional indices is part of the larger topic, laws of indices. It may be useful to explore the main topic before looking into the detailed individual lessons below:

For example here we have a base number of 8 that has been raised to a fractional power

\[8^{\frac{2}{3}} \]

As the denominator is 3 we have to find the cube root of 8 .

\[\sqrt[3]{8} = 2\]

Then, as the numerator is 2 we then square the answer.

\[2^{2} = 4\]

So,

\[\begin{aligned}
8^{\frac{2}{3}} &=(\sqrt[3]{8})^{2} \\
&=4
\end{aligned}\]

**A value raised to the power of ½ means take the square root.**

\[x^{\frac{1}{2}}=\sqrt{x}\]

E.g

\[9^{\frac{1}{2}}=\sqrt{9}=\pm{3}\]

**A value raised to the power of ⅓ means take the cube root**

\[x^{\frac{1}{3}}=\sqrt[3]{x}\]

E.g

\[27^{\frac{1}{3}}=\sqrt[3]{27}=3\]

**A value raised to the power of 4 means take the fourth root.**

\[x^{\frac{1}{4}}=\sqrt[4]{x}\]

E.g

\[16^{\frac{1}{4}}=\sqrt[4]{16}=\pm{2}\]

Etc.

Simplify

\[a^{\frac{1}{4}}\]

**Use the denominator to find the root of the number or letter.**

\[\sqrt[4]{a}\]

2 **Raise the answer to the power of the numerator.**

In this case the numerator is 1 so the answer stays the same

\[\sqrt[4]{a}\]

Evaluate

\[8^{\frac{2}{3}}\]

**Use the denominator to find the root of the number or letter.**

\[\sqrt[3]{8}=2\]

**Raise the answer to the power of the numerator.**

\[2^{2}=4\]

So,

\[8^{\frac{2}{3}}=4\]

This example uses negative numbers as the indices. It is a good idea to check our Laws of Indices page for more information before attempting this question.

Evaluate

\[4^{-\frac{3}{2}}\]

First we need to make the index positive by writing the reciprocal.

\[4^{-\frac{3}{2}}=\frac{1}{4^{\frac{3}{2}}}\]

Then continue to use the steps, focusing on the denominator.

**Use the denominator to find the root of the number or letter.**

\[\sqrt[2]{4}=2\]

**Raise the answer to the power of the numerator.**

\[2^{3}=8\]

So,

\[4^{-\frac{3}{2}}=\frac{1}{8}\]

**Confusing integer and fractional powers**

Raising a term to the power of 2 means we square it

E.g

\[a^{2}=a \times a\]

Raising a term to the power of ½ means we find the square root of it

E.g

\[a^{\frac{1}{2}}=\pm \sqrt{a}\]

Raising a term to the power of 3 means we cube it

E.g

\[a^{3}=a \times a \times a\]

Raising a term to the power of ⅓ means we find the cube root of it

E.g

\[a^{\frac{1}{3}}=\sqrt[3]{a}\]

**Indices, powers or exponents**

Indices can also be called powers or exponents.

1. Evaluate. Give your answer as an integer or fraction.

64^{\frac{1}{2}}

\pm \; 128

\pm \; 8

\pm \; 32

\pm \; 4

The index number tells us to find the square root, so

64^{\frac{1}{2}}=\sqrt{64}=8

2. Evaluate. Give your answer as an integer or fraction.

18

9

3

54

Looking at the index number, the denominator tells us to cube root, and the numerator tells us to square, therefore

\begin{aligned} &27^{\frac{2}{3}}\\ &=(\sqrt[3]{27})^{2}\\ &=3^{2}\\ &=9 \end{aligned}

3. Evaluate. Give your answer as an integer or fraction.

\frac{1}{8}

\frac{1}{64}

\frac{1}{512}

\frac{1}{16}

Looking at the index number, the denominator tells us to square root, the numerator tells us to cube, and as the index is negative we find the reciprocal, therefore

\begin{aligned} &64^{-\frac{3}{2}}\\ &=(\sqrt[2]{64})^{-3}\\ &=\frac{1}{8^{3}}\\ &=\frac{1}{512} \end{aligned}

4. Evaluate

9

27

3

243

Looking at the index number, the denominator tells us to take the fourth root, and the numerator tells us to cube, therefore

\begin{aligned} &81^{\frac{3}{4}}\\ &=(\sqrt[4]{81})^{3}\\ &=3^{3}\\ &=27 \end{aligned}

1. Evaluate

9^{\frac{1}{2}}

**(1 mark)**

Show answer

\pm 3

** (1)**

2. Evaluate

(27)^{\frac{2}{3}}

**(2 marks)**

Show answer

(\sqrt[3]{27})^{2} \\

**(1)**

9

**(1)**

3. Evaluate

\left(\frac{16}{81}\right)^{-\frac{3}{4}}

**(3 marks)**

Show answer

\left(\sqrt[4]{\frac{81}{16}}\right)^{3} \\

**(1)**

**(1)**

\frac{27}{8}=3.375

**(1)**

You have now learned how to:

- Simplify expressions involving the laws of indices
- Calculate with roots, and with integer and fractional indices

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