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Laws of indices Squares and square roots Cubes and cube roots Nth term of a sequence Standard form BIDMAS Fractions Negative numbersThis topic is relevant for:

Here we will learn about expressing a power with a different base.

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.

**Expressing a power with a different base** is simplifying terms involving indices that have different bases.

For example,

Simplify 4\times2^{3} .

As 4=2^{2} , we can substitute this into the expression 4\times2^{3} to get 2^{2}\times 2^{3} .

4\times 2^{3} = 2^{2}\times 2^{3}= 2^5So by expressing a power with a different base, we can simplify the expression.

The key law of indices that we use to change the base is (x^{a})^{b}=x^{a\times b}=x^{ab} .

In order to express a power with a different base:

**Look for a link between the base numbers.****Convert the base to a different power.****Substitute the new base and power into the expression and simplify**.

Get your free expressing a power with a different base worksheet of 20+ powers and roots questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free expressing a power with a different base worksheet of 20+ powers and roots questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREESimplify the expression 3^{4}\times 9^{2} .

**Look for a link between the base numbers.**

3 and 9 are both powers of 3 .

2**Convert the base to a different power.**

As 9=3^{2} , we can use this to change the base of 9 to a base of 3 .

3**Substitute the new base and power into the expression and simplify.**

As 9=3^{2} , we can now say that 3^{4}\times 9^{2} is the same as 3^{4}\times (3^{2})^{2} .

Simplifying this we get 3^{4}\times(3^{2})^{2}=3^{4}\times3^{2\times2}=3^{4}\times3^{4}=3^{4+4}=3^{8} .

So 3^{4}\times9^{2}=3^{8} .

Simplify the following expression into the form a^{b}x^{b} where a and b are integers.

(4x^2)^{3}\div2x .

**Look for a link between the base numbers.**

We can just look at the coefficients of x for this example.

4 is equal to 2^2 we can substitute this into the expression

**Convert the base to a different power.**

4=2^{2}

**Substitute the new base and power into the expression and simplify.**

As 4=2^{2} , we can substitute this into the expression to get,

(4x^2)^{3}\div2x=((2^2)x^2)^{3}\div2x=2^{6}x^{6}\div2x=2^{5}x^{5} .

So (4x^2)^{3}\div2x=2^{5}x^{5} .

Evaluate the expression \frac{8^{3}\times{4^{-1}}}{16} .

**Look for a link between the base numbers.**

4, 8 and 16 are all different powers of 2 and so we can convert them all to simplify the expression.

**Convert the base to a different power.**

8=2^{3}
4=2^{2}
16=2^{4}

**Substitute the new base and power into the expression and simplify.**

We can substitute these into the expression to get

\frac{8^{3}\times{4^{-1}}}{16}=\frac{(2^{3})^{3}\times{(2^{2})^{-1}}}{(2^{4})} =\frac{2^{3\times{3}}\times{2^{2\times{-1}}}}{2^{4}} =\frac{2^{9}\times{2^{-2}}}{2^{4}}=2^{9+-2-4}=2^{9-2-4}=2^{3} .

As the question asks us to evaluate the expression, we get \frac{8^{3}\times{4^{-1}}}{16}=2^{3}=8 .

Express the following as a single power of 5 .

125^{-\frac{2}{3}}**Look for a link between the base numbers.**

We only have one term however we have been told to express this as a power of 5 and so we can change 125 to be a power of 5 .

**Convert the base to a different power.**

125=5^{3}

**Substitute the new base and power into the expression and simplify.**

We can substitute these into the expression to get

125^{-\frac{2}{3}}=(5^{3})^{-\frac{2}{3}}=5^{3\times{-\frac{2}{3}}}=5^{-2} .

Given that y=x^{2} and x^{3}y^{2}=128 , calculate the possible values of x .

**Look for a link between the base numbers.**

We can convert y to be written in terms of x as y=x^{2} .

**Convert the base to a different power.**

We do not need to fulfil this step as this conversion is already given in the question.

**Substitute the new base and power into the expression and simplify.**

We can substitute these into the expression to get

x^3(x^2)^2=x^3\times(x^2)^2=x^3\times x^{2\times2}=x^3\times x^4=x^7 .

Now we have x^{7}=128 and so calculating the seventh root of 128 , we get \sqrt[7]{128}=2 .

So x=2 .

Evaluate \sqrt{3^{4}}+27^{\frac{2}{3}} .

**Look for a link between the base numbers.**

We can convert 27 to be a power of 3 .

**Convert the base to a different power.**

27=3^{3}

**Substitute the new base and power into the expression and simplify.**

We can substitute these into the expression to get

\sqrt{3^{4}}+27^{\frac{2}{3}}=\sqrt{3^{4}}+(3^{3})^{\frac{2}{3}}=\sqrt{3^{4}}+3^{3\times{\frac{2}{3}}}=\sqrt{3^{4}}+3^{2} .

As, \sqrt{3^{4}}=3^{\frac{4}{2}}=3^{2} , we can simplify further to get

\sqrt{3^{4}}=27^{\frac{2}{3}}=3^{2}+3^{2}=9+9=18 .

**Incorrect base**

A possible error is to alter base number without altering the power.

For example,

When converting (4^{2})^{3} to a power of 2 , the base number 4 is changed to a 2 and is incorrectly substituted as (2^{2})^3 .

The correct answer is ((2^{2})^{2})^{3} .

Using brackets can help to avoid this mistake.

**Negative powers and negative number**

A common error is to think that a negative power suggests that the entire number is a negative number. This is incorrect because a negative power means that we are finding a positive reciprocal of the base.

For example, 2^{-4}=-2^{4}=-16 is incorrect.

The correct answer is 2^{-4}=\frac{1}{2^{4}}=\frac{1}{16} .

**Changing the base incorrectly**

For example, 5^{3}\times2^{-4}=(5\times{2})^{3-4}=10^{-1}=\frac{1}{10} . This is incorrect as the bases have been multiplied together. The base must be the same when simplifying calculations with powers. The correct answer would be 5^{3}\times{2^{-4}}=5^{3}\times{\frac{1}{16}}=\frac{125}{16} .

**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} .

1. Express the following as a single power of 5 , 5^{3}\times 25^{2} .

5^{5}

5^{6}

5^{12}

5^{7}

5^{3}\times 25^{2}=5^{3}\times (5^{2})^{2}=5^{3}\times 5^{4}=5^{7}

2. Which expression is equivalent to 8y^{3}(2y)^{2} ?

(16y)^{5}

(2y)^{6}

(2y)^{5}

32y^{6}

8y^{3}=(2y)^{3}

(2y)^{3}\times (2y)^{2}=(2y)^{3+2}=(2y)^{5}

3. Express the following as a single power of 3 , \frac{81^{2}\times{3^{-4}}}{27} .

3^{-1}

3

3^{0}

3^{-3}

\begin{aligned}
&\frac{81^{2} \times 3^{-4}}{27}=\frac{\left(3^{4}\right)^{2} \times 3^{-4}} {3^{3}}=\frac{3^{8} \times 3^{-4}}{3^{3}} \\\\
&=3^{8+-4-3}=3^{8-4-3}=3^{1}=3
\end{aligned}

4. Evaluate 81^{\frac{3}{4}} .

60.75

27

9

3

81^{\frac{3}{4}}=(3^{4})^{\frac{3}{4}}=3^{4\times{\frac{3}{4}}}=3^{3}=27

5. Given that b=a^{-4} and a^{6}b=16 , calculate the value of b .

\frac{1}{8}

-64

2

\frac{1}{256}

a^{6}b=a^{6}\times a^{-4}=a^{2}=16 so a=4 .

b=a^{-4}=\frac{1}{a^{4}}=\frac{1}{4^{4}}=\frac{1}{256} .

6. Evaluate (\frac{9}{25})^{\frac{1}{2}}\times{5^{3}} .

9

75

30

150

(\frac{9}{25})^{\frac{1}{2}}\times{5^{3}}=(\frac{3^{2}}{5^{2}})^{\frac{1}{2}}\times{5^{3}}=\frac{3}{5}\times{5^{3}}=3\times{5^{2}}

=3\times 25=75

1. Circle the expressions that are equivalent to 2^{4} .

4^{2} \hspace{3mm} \sqrt{64} \hspace{3mm} 8^{\frac{4}{3}} \hspace{3mm} 8 ^{-2} \hspace{3mm} \frac{2^{5}\times{2^{7}}}{2^{3}}

**(2 marks)**

Show answer

4^{2} \hspace{3mm} 8^{\frac{4}{3}}

**(2)**

2. (a) Write the number x=3\times 100^{3} in standard form.

(b) Hence find the value of y=2x^{2} . Write your answer in standard form.

**(4 marks)**

Show answer

(a)

x=3\times 10^{6}**(1)**

(b)

x^{2}=(3\times 10^{6})^{2}=9\times 10^{12}**(1)**

**(1)**

**(1)**

3. (a) Express 18 as a product of prime factors in index form.

(b) Use your answer to part a) and laws of indices to write the expression below as a single power of 3 .

\frac{18\times{9^{-4}}}{2\times{3^{6}}}

**(5 marks)**

Show answer

(a)

18=2\times 3^{2}**(1)**

(b)

\frac{2\times{3^{2}}\times{9^{-4}}}{2\times{3^{6}}}**(1)**

**(1)**

**(1)**

**(1)**

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 include powers and roots
- Calculate with roots, and with integer {and fractional} indices
- Simplify expressions involving sums, products and powers, including the laws of indices

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