Learning Goal

Syllabus Requirement

Students need to learn the laws of indices for all rational exponents, with the below point being highlighted in particular:

•    The equivalence of ??a^{\frac{m}{n}}?? and ??\sqrt[n]{a^{m}}??.

Study Notes

There are 7 basic laws of indices that should be learnt and understood. They are listed in the information box below:

The 7 basic laws of indices.

Law 1:
??x^{m} \times x^{n} = x^{(m+n)}??

Law 2:
??x^{m} \div x^{n} = x^{(m-n)}??

Law 3:
??(x^{m})^{n} = x^{mn}??

Law 4:
??x^{0} = 1??

Law 5:
??x^{-n} = \frac{1}{x^{n}}??

Law 6:
??x^{\frac{m}{n}} = \sqrt[n]{x^{m}} = (\sqrt[n]{x})^{m}??

Law 7:
??x^{\frac{1}{n}} = \sqrt[n]{x}??

Note that Law 7 is actually just a special case of Law 6 (i.e. the case where ??m = 1??).

It is important to understand two terms, the base is the number, or expression that is being raised to a power or index. For example, in the term ??a^{b}??, the base is ??a?? and the power or index is ??b??.

Worked Examples

We'll now run through some examples of each of these rules in turn.

Law 1

Example 1

Question

Simplify the following expression ?? x^{2} \times x^{3} ??.

Example 2

Question

Simplify the following expression ?? x^{4} \times x^{3} \times x^{5} ??.

Law 2

Example 3

Question

Simplify the following expression ?? x^{3} \div x^{2} ??.

Law 3

Example 4

Question

Simplify the following expression ?? (x^{2})^{5} ??.

Law 4

Example 5

Question

Simplify the following expression ?? x^{3} \times x^{-3} ??.

Law 5

Example 6

Question

Write the following expression with only positive indices ?? x^{-3} \times x^{-3} ??.

Law 6

Example 7

Question

Simplify the following expression ?? x^{\frac{4}{3}}??.

Law 7

Example 8

Question

Simplify the following expression ?? x^{\frac{1}{3}}??.

Exam Questions

The table below contains every exam question that has been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

Question
Exam Board Subject Paper Year Month Module Question
No.
Parts Total
Marks
Exam
Paper
Mark
Scheme
Edexcel Maths Standard 2005 January Core 1 1 (a), (b) 3
Edexcel Maths Standard 2005 June Core 1 1 (a), (b) 3
Edexcel Maths Standard 2007 June Core 1 2 (a), (b) 4
Edexcel Maths Standard 2008 January Core 1 2 (a), (b) 3
Edexcel Maths Standard 2009 January Core 1 1 (a), (b) 3
Edexcel Maths Standard 2009 June Core 1 2 n/a 3
Edexcel Maths Standard 2011 January Core 1 1 (a), (b) 4
Edexcel Maths Standard 2011 June Core 1 1 (a), (b) 3
Edexcel Maths Standard 2012 June Core 1 2 (a), (b) 4
Edexcel Maths Standard 2013 January Core 1 2 n/a 2
Edexcel Maths Standard 2013 June Core 1 3 (a), (b) 5
Edexcel Maths International 2013 June Core 1 5 (a), (b) 5

Exam Tips

1. Exam questions are often simple applications of the above laws, for example, often taking the form of evaluating expressions such as:
??25^{\frac{3}{2}}??
??100^{\frac{1}{2}}??

2. Sometimes exam questions have focused on rewriting an expression in another form, for example:
Rewrite ??8^{3}?? as ??2^{x}??
Rewrite ??16^{\frac{3}{2}}?? as ??4^{x}??

3. Remember, there is no calculator allowed, so the examiners won’t ask you to complete complicated arithmetic. Consider what order you do any calculation in to make the arithmetic easier. For the questions below, the easy way is shown in green and the hard way in red.
Find ??25^{\frac{3}{2}}??.
??25^{\frac{3}{2}} = \sqrt{25 \times 25 \times 25} = \sqrt{15625} = 125??.
??25^{\frac{3}{2}} = \sqrt{25} \times \sqrt{25} \times \sqrt{25} = 5 \times 5 \times 5 = 125??.
Find ??27^{\frac{4}{3}}??.
??27^{\frac{4}{3}} = \sqrt[3]{27^{4}} = \sqrt[3]{27 \times 27 \times 27 \times 27} = =\sqrt[3]{531441} = 81??.
??27^{\frac{4}{3}} = \sqrt[3]{27} \times \sqrt[3]{27} \times \sqrt[3]{27} \times \sqrt[3]{27} = 3 \times 3 \times 3 \times 3 = 81??.

4. You should learn and become familiar with the laws above. None of the information from this page is in the materials given to you in the exam ('Mathematical Formulae') and the methods used here will often be required in other questions, often, for example, in questions about surds.