Algebraic Roots & Indices (Edexcel IGCSE Maths)

Exam board Pearson Edexcel · IGCSEUpdated 2026-01-26

Spec point — 1.4C

Algebraic roots & indices

What are the laws of indices?

Index laws are rules you can use when doing operations with powers
- They work with both numbers and algebra
Consider the term $a^{n}$
- $a$ is the base number
- $n$ is the index

Law	Description	How it works
$a^{1}=a$	Anything to the power of 1 is itself	$x^{1}=x$
$a^{0}=1$	Anything to the power of 0 is 1	$b^{0}=1$
$a^{m}\times a^{n}=a^{m+n}$	To multiply indices with the same base, add their powers	$c^{3} \times c^{2} \\ = ( c \times c \times c ) \times ( c \times c ) \\ = c^{5}$
$a^{m}\div a^{n}=\frac{a^{m}}{a^{n}}=a^{m-n}$	To divide indices with the same base, subtract their powers	$d^{5 \div d^{2} \\ = \frac{d \times d \times d \times \cancel{d} \times \cancel{d} }{ \cancel{d} \times \cancel{d} } \\ = d^{3} }$
$( a^{m )^{n} = a^{m} n}$	To raise indices to a new power, multiply their powers	$( e^{3} )^{2} \\ = ( e \times e \times e ) \times ( e \times e \times e ) \\ = e^{6}$
$( a b )^{n} = a^{n} b^{n}$	To raise a product to a power, apply the power to both numbers, and multiply	$( f \times g )^{2} \\ = f^{2} \times g^{2} \\ = f^{2} g^{2}$
`( a / b ) ^ n = a ^ n / b ^ n`	To raise a fraction to a power, apply the power to both the numerator and denominator	$( \frac{h}{i} )^{2} = \frac{h^{2}}{i^{2}}$
$a^{-1}=\frac{1}{a}$ $a^{-n}=\frac{1}{a^{n}}$	A negative power is the reciprocal	$j^{-1}=\frac{1}{j}$ $k^{-3}=\frac{1}{k^{3}}$

These can be used to simplify expressions
- Work out the number and algebra parts separately
  - $( 3 x^{7 ) \times ( 6 x^{4} ) = ( 3 \times 6 ) \times ( x^{7} \times x^{4} ) = 18 x^{7} + 4} = 18 x^{11}$
  - $\frac{6x^{7}}{3x^{4}}=\frac{6}{3}\times\frac{x^{7}}{x^{4}}=2x^{7-4}=2x^{3}$
  - $( 3 x^{7} )^{2} = ( 3 )^{2} \times ( x^{7} )^{2} = 9 x^{14}$

How can I solve equations with an unknown in the index?

Write both sides of the equation over the same base number
- Then work out what x should be $\begin{gathered} 5^{x} & = 125 \\ 5^{x} & = 5^{3} \\ x & = 3 \end{gathered}$
You might have to use negative indices $\begin{array}{rcl} 2^{x & = \frac{1}{8} \\ 2^{x} & = \frac{1}{2^{3}} \\ 2^{x} & = 2^{-3} } \\ x = - 3 \end{array}$

Worked Example (a) Simplify $( u^{5} )^{5}$

Answer:

Use $( a^{m )^{n} = a^{m} n}$

$( u^{5 )^{5} = u^{5} \times 5}$

$u^{25}$

(b) If $q^{x}=\frac{q^{2}\times q^{5}}{q^{10}}$ find $x$ .

Answer:

Use $a^{m}\times a^{n}=a^{m+n}$ to simplify the numerator

$q^{2}\times q^{5}=q^{2+5}=q^{7}$

Use $\frac{a^{m}}{a^{n}}=a^{m-n}$ * *to simplify the fraction

$\frac{q^{7}}{q^{10}}=q^{7-10}=q^{-3}$

Write out both sides of the equation

$q^{x}=q^{-3}$ * *

Both sides are now over the same base of $q$

So $x$ must equal the power on the right-hand side

$x=-3$

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Law

Description

How it works

a^{1}=a

Anything to the power of 1 is itself

x^{1}=x

a^{0}=1

Anything to the power of 0 is 1

b^{0}=1

a^{m}\times a^{n}=a^{m+n}

To multiply indices with the same base, add their powers

c^{3} \times c^{2} \\ = ( c \times c \times c ) \times ( c \times c ) \\ = c^{5}

a^{m}\div a^{n}=\frac{a^{m}}{a^{n}}=a^{m-n}

To divide indices with the same base, subtract their powers

d^{5 \div d^{2} \\ = \frac{d \times d \times d \times \cancel{d} \times \cancel{d} }{ \cancel{d} \times \cancel{d} } \\ = d^{3} }

( a^{m )^{n} = a^{m} n}

To raise indices to a new power, multiply their powers

( e^{3} )^{2} \\ = ( e \times e \times e ) \times ( e \times e \times e ) \\ = e^{6}

( a b )^{n} = a^{n} b^{n}

To raise a product to a power, apply the power to both numbers, and multiply

( f \times g )^{2} \\ = f^{2} \times g^{2} \\ = f^{2} g^{2}

( a / b ) ^ n = a ^ n / b ^ n

To raise a fraction to a power, apply the power to both the numerator and denominator

( \frac{h}{i} )^{2} = \frac{h^{2}}{i^{2}}

a^{-1}=\frac{1}{a}

a^{-n}=\frac{1}{a^{n}}

A negative power is the reciprocal

j^{-1}=\frac{1}{j}

k^{-3}=\frac{1}{k^{3}}