Algebraic Roots & Indices (Cambridge (CIE) IGCSE Maths): Revision Note

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

Law

Description

How it works

a to the power of 1 equals a

Anything to the power of 1 is itself

x to the power of 1 equals x

a to the power of 0 equals 1

Anything to the power of 0 is 1

b to the power of 0 equals 1

a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent

To multiply indices with the same base, add their powers

c cubed cross times c squared
equals open parentheses c cross times c cross times c close parentheses cross times open parentheses c cross times c close parentheses
equals c to the power of 5

a to the power of m divided by a to the power of n equals a to the power of m over a to the power of n equals a to the power of m minus n end exponent

To divide indices with the same base, subtract their powers

d to the power of 5 divided by d squared
equals fraction numerator d cross times d cross times d cross times up diagonal strike d cross times up diagonal strike d over denominator up diagonal strike d cross times up diagonal strike d end fraction
equals d to the power of 3 space end exponent

open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent

To raise indices to a new power, multiply their powers

open parentheses e cubed close parentheses squared
equals open parentheses e cross times e cross times e close parentheses cross times open parentheses e cross times e cross times e close parentheses
equals e to the power of 6

open parentheses a b close parentheses to the power of n equals a to the power of n b to the power of n

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

open parentheses f cross times g close parentheses squared
equals f squared cross times g squared
equals f squared g squared

open parentheses a over b close parentheses to the power of n equals a to the power of n over b to the power of n

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

open parentheses h over i close parentheses squared equals h squared over i squared

a to the power of negative 1 end exponent equals 1 over a

a to the power of negative n end exponent equals 1 over a to the power of n

A negative power is the reciprocal

space j to the power of negative 1 end exponent equals 1 over j

k to the power of negative 3 end exponent equals 1 over k cubed

  • These can be used to simplify expressions 

    • Work out the number and algebra parts separately

      • open parentheses 3 x to the power of 7 close parentheses cross times open parentheses 6 x to the power of 4 close parentheses equals open parentheses 3 cross times 6 close parentheses cross times open parentheses x to the power of 7 cross times x to the power of 4 close parentheses equals 18 x to the power of 7 plus 4 end exponent equals 18 x to the power of 11

      • fraction numerator 6 x to the power of 7 over denominator 3 x to the power of 4 end fraction equals 6 over 3 cross times x to the power of 7 over x to the power of 4 equals 2 x to the power of 7 minus 4 end exponent equals 2 x to the power of 3 space end exponent

      • open parentheses 3 x to the power of 7 close parentheses squared equals open parentheses 3 close parentheses squared cross times open parentheses x to the power of 7 close parentheses squared equals 9 x to the power of 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
      table row cell 5 to the power of x end cell equals 125 row cell 5 to the power of x end cell equals cell 5 cubed end cell row cell x space end cell equals cell space 3 end cell end table

  • You might have to use negative indices
    table attributes columnalign right center left columnspacing 0px end attributes row cell 2 to the power of x end cell equals cell 1 over 8 end cell row cell 2 to the power of x end cell equals cell 1 over 2 cubed end cell row cell 2 to the power of x end cell equals cell 2 to the power of negative 3 end exponent end cell row x equals cell negative 3 end cell end table

Worked Example

(a) Simplify open parentheses u to the power of 5 close parentheses to the power of 5

 Use open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent

open parentheses u to the power of 5 close parentheses to the power of 5 equals u to the power of 5 cross times 5 end exponent

bold italic u to the power of bold 25

(b) If  q to the power of x equals fraction numerator q squared cross times q to the power of 5 over denominator q to the power of 10 end fraction   find x.

Use a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent to simplify the numerator

q squared cross times q to the power of 5 equals q to the power of 2 plus 5 end exponent equals q to the power of 7

Use a to the power of m over a to the power of n equals a to the power of m minus n end exponent to simplify the fraction

q to the power of 7 over q to the power of 10 equals q to the power of 7 minus 10 end exponent equals q to the power of negative 3 end exponent

Write out both sides of the equation

q to the power of x equals q to the power of negative 3 end exponent 

Both sides are now over the same base of q

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

bold italic x bold equals bold minus bold 3

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