Properties of Indices (Edexcel IGCSE Further Pure Maths): Revision Note

Laws of Indices

What laws of indices do I need to know?

• In an expression like $a^n$ 

  • $a$ is known as the base

  • $n$ is known as the index (also called the power or exponent)

• The index laws you need to know and be able to use are summarised here:

  • $a^m\times a^n=a^{m+n}$

  • $a^m÷a^{\mathit{n}}=a^{\mathit{m}\mathit{-}\mathit{n}}$

    • Be careful!  The two laws above can only be used if the two terms on the left-hand side of the equation have the same base.

  • ${\left(a^m\right)}^n=a^{mn}$

  • ${\left(ab\right)}^n=a^n{b}^n$

  • $a^1=a$

  • $a^0=1$

  • $a^{\frac{1}{m}}=\sqrt[m]{a}$

  • $a^{\frac{m}{n}}=\sqrt[n]{a^m}={\left(\sqrt[n]{a}\right)}^m$

  • $a^{-m}=\frac{1}{a^m}$

How do I work with laws of indices?

• Laws of indices work with numerical and algebraic terms

• These can be used to simplify expressions where terms are multiplied or divided

  • Deal with the number and algebraic parts separately

    • $\left(3x^7\right)\times \left(6x^4\right)=\left(3\times 6\right)\times \left(x^7\times x^4\right)=18x^{11}$

    • $\frac{3x^7}{6x^4}=\frac{3}{6}\times \frac{x^7}{x^4}=\frac{1}{2}{x}^3$

    • ${\left(3x^7\right)}^2={\left(3\right)}^2\times {\left(x^7\right)}^2=9x^{14}$

How can I solve equations when the unknown is in the index?

• If two terms with indices are equal and the terms have the same positive base (other than 1) then the indices must be equal

  • If $a^x=a^y$ then  $x=y$

    • Not valid if $a\le 0$ or $a=1$

• If the unknown is part of the index then write both sides with the same base number

  • Then ignore the base number, make the indices equal and solve that equation

$\begin{array}{rcl}{5}^{2x}& =& 125\\ 5^{2x}& =& 5^3\\ 2x& =& 3\\ x& =& \frac{3}{2}\end{array}$

• In more complicated questions you might have to use negative and fractional indices

  • You may also have to rewrite both sides with the same base number

$\begin{array}{rcl}{8}^x& =& \frac{1}{4}\\ {\left(2^3\right)}^x& =& \frac{1}{2^2}\\ 2^{3x}& =& 2^{-2}\\ 3x& =& -2\\ x& =& -\frac{2}{3}\end{array}$