A Level Maths: Indices
Introduction
Here we are going to recap the rules of indices that you have met whilst doing GCSE Maths. You will learn how to simplify expressions involving indices including fractional and negative powers.
There are some additional articles on the site regarding index laws and some examples of how they are applied at A Level Maths, but given the fact that this topic appears in many other topics especially calculus then you need to be more than an expert at this topic.
Terminology
The expression a^m consists of a base a and m is the index or power in which the base is raised.
Multiplication Rule
If you have 2^3 \times 2^2 this gives 2 \times 2 \times 2 \times 2 \times 2=2^5
It is important to note that there is no need to show all the 2’s. A much simpler method is to add the powers i.e. 3+2=5.
So 2^3 \times 2^2=2^{3+2}=2^5
In general this can be written as: a^m \times a^n=a^{m+n}
Raising a base to a power and to another power
Consider \left(2^2\right)^2 which can be written as \left(2^2\right) \times\left(2^2\right)=2 \times 2 \times 2 \times 2=2^4
A much simpler approach is to simply multiply the powers together.
In general this can be written as: \left(a^m\right)^n=a^{m n}
Division Rule
Consider 3^5 \div 3^2 which can be calculated as follows:
\frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}=3 \times 3 \times 3=3^3A simpler approach would have been to have subtracted the powers, 5 – 2 = 3.
So 3^5 \div 3^2=3^{5-2}=3^3.
In general this can be written as: a^m \div a^n=a^{m-n}
The power of zero
If you divide 3^2 by 3^2the answer will be 1.
Following the rule for division of indices, 3^2 \div 3^2=3^0=1
It therefore follows that anything that is raised to the power 0 will always be equal to 1.
In general this can be written as: a^0=1
Negative indices
Consider \begin{aligned} 4^2 \div 4^4=\frac{4 \times 4}{4 \times 4 \times 4 \times 4} & \\ 4^{2-4} & =\frac{1}{4 \times 4} \\ 4^{-2} & =\frac{1}{4^2} \end{aligned}
In general it can be written that: a^{-n}=\frac{1}{a^n}
Example
Simply \left(4 \sqrt{2} \times \frac{1}{16} \times \sqrt[5]{32}\right)^2
Solution
With a question like this your aim is to put each term in terms of the same base and ideally to use the lowest base.
4=2^2 ; \frac{1}{16}=\frac{1}{2^4}=2^{-4} ; 32=2^5So a common base here would be 2.
\begin{aligned} \left(4 \sqrt{2} \times \frac{1}{16} \times \sqrt[5]{32}\right)^2 & =\left(2 \times 2^{\frac{1}{2}} \times 2^{-4} \times\left(2^5\right)^{\frac{1}{5}}\right)^2 \\ = & \left(2^{\frac{5}{2}} \times 2^{-4} \times 2\right)^2 \\ = & \left(2^{-\frac{1}{2}}\right)^2 \\ & =2^{-1} \\ & =\frac{1}{2} \end{aligned}From the outset this question did not look like it was going to be straightforward. But by breaking the numbers down into a common base, the question has become manageable. Quite often during your A Level Maths course you are having to break numbers down and you are essentially working backwards from what you were used to doing at GCSE.
Some of the questions do involve the skill of factorising and what you have learnt at GCSE you need to retain for A Level maths. If you think you need the aid of a UK based online maths tutor then you should contact us immediately and we can put you in touch with an experienced tutor.
Example
Simplify 3(x+7)^{\frac{1}{2}}-\frac{2}{3}(x+7)^{\frac{3}{2}}
Solution
Here the common term will be (x+7)^{\frac{1}{2}} and the above can be written as:
\begin{gathered} 3(x+7)^{\frac{1}{2}}-\frac{2}{3}(x+7)^{\frac{1}{2}}(x+7) \\ =(x+7)^{\frac{1}{2}}\left[3-\frac{2}{3}(x+7)\right] \\ =(x+7)^{\frac{1}{2}}\left[\frac{9-2 x-14}{3}\right] \\ =(x+7)^{\frac{1}{2}} \frac{(-5-2 x)}{3} \\ =-\frac{(5+2 x)}{3} \sqrt{(x+7)} \end{gathered}The last two entries that are highlighted are both correct answers. The last entry is simply an alternative answer in terms of how the answer is given.
The last few questions have shown how indices rules are used in a different light to what you have seen at GCSE. At A Level Maths this type of skill is expected and it is something that cannot be done with a calculator.
As well as using your skills in indices the last few questions also saw factorising being used which again a skill that you need to be confident in as quite often a question will tell you to factorise.
Your A Level Maths Revision needs to consist of doing questions like these. And just be careful how you format your answer. If a question does not ask for an answer in a particular form then when you check your answer you might see the answer being different. It does not mean that you are wrong. The answer could just be written differently.
Here are some questions for you to try and take extra care with question 10 that requires you to factorise.