GCSE Maths Number

Powers And Roots

Fractional Powers

# Fractional Powers

Here we will learn about fractional powers, including what they are, how to simplify and evaluate them, and how to combine them with other laws of exponents.

There are also powers and roots worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are fractional powers?

Fractional powers are a type of index that represents the nth root of a number.

In general, the rule for fractional exponents is x^{\frac{m}{n}}=\sqrt[n]{x^{m}} .

Where m and n are constants and n 0 .

For example,

9^{\frac{1}{2}}=\sqrt{9}=3

m=1, n=2, and x=9 , so we are calculating the second root, or the square root of 9 which is equal to 3 .

## How to use fractional powers

In order to use fractional powers:

1. Simplify any powers using laws of indices.
2. Evaluate or solve (if required).

## Fractional powers examples

### Example 1: simplify and evaluate

Simplify and evaluate the expression

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

1. Simplify any powers using laws of indices.

As x^{a} \times x^{b}=x^{a+b} , and 9=9^{1} , we have

9^{\frac{1}{2}}=9^{\frac{1}{2}+1}=9^{1.5}=9^{\frac{3}{2}} .

2Evaluate or solve (if required).

Evaluating the expression, we get

9^{\frac{3}{2}}=(\sqrt{9})^{3}=3^{3}=27 .

So 9^{\frac{1}{2}}\times\ {9}=27 .

### Example 2: algebraic

Write the following expression in its simplest form

(xy)^{a}\div\ {(x)^\frac{a}{2}} .

As we have multiple variables in the first expression raised to the power of a , let’s remove the brackets so we know the power of each variable separately,

(xy)^{a}=x^{a}\times y^{a} .

We can then divide x^{a} \times y^{a} by (x)^\frac{a}{2} to get

x^{a}\times\ {y^{a}}\div{(x)^\frac{a}{2}}=x^{a-\frac{a}{2}}\times\ {y^{a}}=x^{\frac{a}{2}}\times\ {y^{a}} .

(Remember we can only simplify powers if the base is the same.)

So we now have x^{\frac{a}{2}}\times\ {y^{a}} .

The question asks us to simplify the expression and so the solution is,

x^{\frac{a}{2}}y^{a} .

### Example 3: inequalities

Solve the inequality

64^{\frac{1}{3}}\geq{\frac{1}{x}} .

Looking at the left hand side of the inequality sign, we have 64 being raised to the power of \frac{1}{3} and so we can write this as

64^{\frac{1}{3}}=\sqrt[3]{64}=4 .

So we now have the inequality 4\geq{\frac{1}{x}} .

The inverse for dividing by x is multiplying, so we need to multiply both sides by x .

4x\geq 1

This can be simplified by dividing both sides of the inequality by 4 ,

x\geq{\frac{1}{4}} .

This is the solution.

### Example 4: negative fractional exponents

Evaluate the following expression,

216^{-\frac{2}{3}} .

As the power is a fraction and a negative number, we are going to simplify the base by looking at the denominator of the power first.

As the denominator of the fraction is the nth root of the base number we can start to simplify the expression,

216^{-\frac{2}{3}}=(216^{\frac{1}{3}})^{-2}=(\sqrt[3]{216})^{-2}=6^{-2} .

(The power of \frac{1}{3} is exactly the same as the cube root.)

6^{-2}=\frac{1}{6^{2}}

The expression \frac{1}{6^{2}} is the simplified form of the expression, but we can now evaluate the expression as we know that 6^{2}=36 so \frac{1}{6^{2}}=\frac{1}{36} .

So the solution is \frac{1}{36} .

### Example 5: fractional base number

Simplify fully,

(2\frac{113}{256})^{\frac{1}{4}} .

Write your answer as a mixed number.

As the base number is a mixed number, let’s convert it to an improper fraction first,

2\frac{113}{256}=\frac{2\times{256}+113}{256}=\frac{625}{256} .

So we now have (\frac{625}{256})^{\frac{1}{4}} .

As the fraction is raised to the power of \frac{1}{4} , both the numerator and the denominator can be raised to the power of \frac{1}{4} .

(\frac{625}{256})^{\frac{1}{4}}=\frac{625^{\frac{1}{4}}}{256^{\frac{1}{4}}}

As the index is equal to \frac{1}{4} , we can calculate the 4^{th} root of each part of the fraction.

The fourth root of 625 is 5 .

The fourth root of 256 is 4 .

So, \frac{\sqrt[4]{625}}{\sqrt[4]{256}}=\frac{5}{4}=1\frac{1}{4} .

So, (2\frac{113}{256})^{\frac{1}{4}}=1\frac{1}{4} .

### Example 6: expressing a power with a different base

Simplify fully,

(8^{-10})^{\frac{1}{5}} .

Write your answer as a fractions in terms of the power of 2 .

The law of indices we are going to use involves parentheses, namely (x^{a})^{b}=x^{a\times b} , we get

(8^{-10})^{\frac{1}{5}}=8^{-10\times{\frac{1}{5}}}=8^{-2} .

8^{-2}=\frac{1}{8^{2}} .

The question does not want us to state the value of \frac{1}{8^{2}} .

Instead, the question asks for the solution to be written as a power of 2 , so we need to change the expression so that the base number is 2 , instead of 8 .

As 8=2^{3} , we have \frac{1}{8^{2}}=\frac{1}{(2^{3})^{2}}=\frac{1}{2^{3\times{2}}}=\frac{1}{2^{6}}

So the solution is \frac{1}{2^{6}} .

### Common misconceptions

• Fractional power and fraction of amounts

A common error is to mistake a fractional power with multiplying by a fraction.

For example,

27^{\frac{1}{3}}=27\times{\frac{1}{3}}=9 is incorrect.

This is because the denominator of the fraction is the nth root of the base.

The correct answer is

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

• BIDMAS

When raising a base to a power, the power is associated with a variable or an expression.

For example, let’s look at  3x^{-2} .

Using BIDMAS we can see that x is being raised to the power of -2 , and then is multiplied by 3 to get the answer \frac{3}{x^{2}}

However a common error is to calculate 3x raised to the power of -2 , giving the answer \frac{1}{9x^{2}} which is incorrect. If this was the case, the question would be written as (3x)^{-2}.

• Incorrect application of the exponent

A similar circumstance is applied to fractions that are raised to a power.

For example, let’s look at (\frac{5}{2})^{-3} .

The correct application of the laws of indices would give a correct answer of  (\frac{5}{2})^{-3}=(\frac{2}{5})^{3}=\frac{8}{125} .

However a common error would be to apply the -3 only to the numerator, leaving the denominator unchanged. This would give the incorrect answer of (\frac{5}{2})^{-3}=\frac{1}{125}\times\frac{1}{2}=\frac{1}{250} .

### Practice fractional powers questions

1. Evaluate

4^{\frac{1}{2}}\times\ {4^{\frac{3}{2}}} .

4^{\frac{3}{4}}

12

8

16
4^{\frac{1}{2}}\times\ {4^{\frac{3}{2}}}=4^{\frac{1}{2}+\frac{3}{2}}=4^{\frac{4}{2}}=4^{2}=16

2. Simplify fully,

(x^{a}y)^{\frac{1}{a}} .

\frac{x^{a}}{y^{a}}

(xy)^{a}

xy^{\frac{1}{a}}

\frac{1}{ax^{a}y}
(x^{a}y)^{\frac{1}{a}}=x^{a\times{\frac{1}{a}}}\times\ {y^{\frac{1}{a}}}=x\times\ {y^{\frac{1}{a}}}=xy^{\frac{1}{a}}

3. State the range of values for the following inequality,

8^{\frac{2}{3}}<\frac{x}{5} .

x>20

x<20

x>30

x<30
8^{\frac{2}{3}}=(\sqrt[3]{8})^{2}=2^{2}=4

4<\frac{x}{5}

20<x so x>20

4. Evaluate

625^{-\frac{3}{4}} .

-125

125

\frac{1}{125}

-\frac{1}{125}
625^{-\frac{3}{4}}=\frac{1}{625^{\frac{3}{4}}}=\frac{1}{(\sqrt[4]{625})^{3}}=\frac{1}{5^{3}}=\frac{1}{125}

5. Evaluate

(\frac{1024}{32})^{\frac{1}{5}} .

\frac{1}{8}

2

6.4

160
(\frac{1024}{32})^{\frac{1}{5}}=\frac{\sqrt[5]{1024}}{\sqrt[5]{32}}=\frac{4}{2}=2

6. Simplify the expression below. Write your answer in the form 3^{n}.

(9^{-3})^{\frac{1}{2}}

-\frac{3}{2}

3^{-3}

\frac{1}{27}

27
(9^{-3})^{\frac{1}{2}}=(9^{\frac{1}{2}})^{-3}=3^{-3}=\frac{1}{3^{3}}=\frac{1}{27}

### Fractional powers GCSE questions

1. Solve the inequality 3\div \sqrt[3]{x}<x^{\frac{2}{3}} .

(4 marks)

2. (a) Evaluate 16^{-1\frac{1}{2}} .

(b) Simplify a^{\frac{1}{2}}\times{a^{\frac{1}{2}}} .

(5 marks)

(a)

-1\frac{1}{2}=-\frac{3}{2}

(1)

\sqrt{16}=4

(1)

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

(1)

(b)

a^{\frac{1}{2}}\times{a^{\frac{1}{2}}}=a^{\frac{1}{2}+\frac{1}{2}}

(1)

a^{1}

(1)

3. Express the following as a single power of 2 ,

4^{4}\times{\sqrt{8}} .

(6 marks)

4^{4}=(2^{2})^{4}

(1)

(2^{2})^{4}=2^{2\times 4}=2^{8}

(1)

\sqrt{8}=8^{\frac{1}{2}}=(2^{3})^{\frac{1}{2}}

(1)

2^{8}\times{2^{\frac{3}{2}}}

(1)

2^{8+\frac{3}{2}}

(1)

=2^{\frac{19}{2}}=2^{9\frac{1}{2}}

(1)

## Learning checklist

You have now learned how to:

• Consolidate your numerical and mathematical capability from key stage 3 and extend your understanding of the number system to fractional indices
• Calculate with roots, and fractional indices

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