Indices

Topic Overview

An index or the plural 'indices' refers to a sequential arrangement of numbers.

In order to understand and use indices, you must first understand the basic principles of squared numbers, cubed numbers, and their roots.

A squared number is displayed as 42 . This means that the number 4 is squared, or 4x4 .

The small 2 is referred to as an index number, or power. It is this small number which tells you how many times you must multiply the main value (in this case 4) by itself.

This series of numbers are known as square numbers. These are the values which you get when you multiply an integer by itself. For example, the first 5 square numbers are as follows:

\[{1^2} = 1\]
\[{2^2} = 4\]
\[{3^2} = 9\]
\[{4^2} = 16\]
\[{5^2} = 25\]

The opposite of a square number is a square root. The square root of a number is often denoted by this symbol:

\[\sqrt {} \]

Therefore if you are asked to work out the square root f 4, it will be denoted by the following symbol:

\[\sqrt 4 = 2\]

However, when you square either a positive or negative value, you will always receive a positive result. For example -2 x-2 = 4.

Therefore,

\[\sqrt 4 = \pm 2\]

It is important to remember that every number has two square roots. As well as squaring a number , you can also cube a number. A cubed number is displayed as 83 and means that the number has been cubed or 8x8x8. Similarly to square numbers, there are also cube numbers. The first 5 cube numbers are as follows:

\[{1^3} = 1\]
\[{2^3} = 8\]
\[{3^3} = 27\]
\[{4^3} = 64\]
\[{5^3} = 125\]

Similarly to square roots, the opposite of a cube number is known as a cube root, which is denoted using the following symbol:

\[\sqrt[3]{{}}\]

Therefore,

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

However, unlike squared numbers, each number only has one cube root.

In the new linear GCSE Maths paper, you will be required to work out several problems involving indices. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

  • Use index notation for squares, cubes and powers of 10
  • Use index laws for multiplication and division of integer, fractional and negative powers
  • Interpret, order and calculate with numbers written in standard index form

Listed below are a series of summaries and worked examples to help you solidify your knowledge about indices.

1 - Rules of Indices
Once you have a basic understanding of squared and cubed numbers, you can perform operations on these values. When multiplying and dividing and working out the power of a power, you must follow the three rules of indices:

Multiplying

When multiplying indices, you add the values of the powers.

Example
(a) - Calculate

\[{3^3}*{3^5}\]

Solution
(a) -

\[{3^3}*{3^5} = (3*3*3)*(3*3*3*3*3)\]
\[3 + 5 = 8\]
\[{3^3}*{3^5} = {3^8}\]
\[{3^8} = 6561\]

Dividing

Alternatively, when dividing indices you subtract the values of the powers.

Example
(b) - Calculate

\[{3^5} + {3^3}\]

Solution
(b) -

\[{3^5}/{3^3} = (3*3*3*3*3)/(3*3*3)\]

Therefore you cancel out two of the 3s because 5-3= 2

Therefore,

\[{3^5}/{3^3} = {3^2} = 9\]

The Power of a Power

When taking the power of a number which has already been raised to a power, you must multiply the values of these powers.

Example
(c) - Calculate

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

Solution
(c) - In this question, you are being asked to calculate the square of

\[{3^3}\]

To calculate this, you multiply the powers :

\[{({3^3})^2} = (3*3*3)*(3*3*3) = {3^6}\]

The answer has an index of 6, which is the result of multiplying the powers at the beginning:

\[3*2 = 6\]

Therefore,

\[{({3^3})^2} = {3^6} = 729\]

2 - Zero, Negative and Fractional Powers
During your exam, you may be asked to work out zero, negative and fractional powers. In order to do so correctly, you must follow the specific rules which apply to these powers:

1 - Anything raised to the power of 0 is equal to one. For example,

\[{1^0} = 1,{2^0} = 1,{3^0} = 1,...\]

2 - Anything raised to the power of a negative value is

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

For example:

\[{2^{ - 2}} = {(\frac{1}{2})^2} = \frac{1}{4}\]
\[{3^{ - 4}} = {(\frac{1}{3})^4} = \frac{1}{{81}}\]

3 - Anything raised to the power of a fractional value is the root of that value or:

\[{a^{1/2}} = \sqrt a \]
\[{a^{1/3}} = \sqrt[3]{a}\]

Example
(a) - Calculate

\[{4^{1/2}}\]

Solution
(a) -

\[{4^{1/2}} = \sqrt 4 = 2\]

Example
(b) - Calculate

\[{27^{1/3}}\]

Solution
(b) -

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

Exam Tips

  1. Remember that when multiplying indices, you add the powers
  2. When dividing indices, remember that you subtract the powers
  3. When taking the power of a number which has already been raised to a power, you must multiply the values of these powers
  4. Any value raised to the power of 0 is equal to 1
  5. Any value raised to the power of a negative value is 1/ a^b
  6. Any value raised to the power of a fractional value is the root of that value

Topic Summary

Ultimately, once you have memorised the fundamental rules of indices, you will be able to easily calculate the answers to any indices related questions during your GCSE maths exam. Also, it is extremely important that you clearly display all of your working out. By doing so, you can earn valuable method marks which can make a significant difference to your overall grade.

Related Topics

  • Arithmetic with Fractions
  • Problem Solving with Decimals and Percentages
  • Accuracy of Measurement Problems
  • Arithmetic with Positive and Negative Integers
  • Standard Form
  • Rational and Irrational Numbers
  • Combinations