GCSEMathsWJECMaths & Numeracy (Double Award)FoundationRevision NotesAlgebraAlgebraic Conventions, Expressions & IndicesAlgebraic Roots & Indices

Algebraic Roots & Indices (WJEC GCSE Maths & Numeracy (Double Award)): Revision Note

Exam code: 3320

Mark Curtis

Written by: Mark Curtis

Reviewed by: Jamie Wood

Updated on

WJEC Maths & Numeracy (Double Award): FoundationWJEC Eduqas Maths: FoundationOCR Maths: FoundationAQA Maths: FoundationEdexcel Maths: Foundation

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 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

  • 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

Worked Example

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

Answer:

 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

u to the power of 25

(b) If  q to the power of x equals fraction numerator q to the power of 12 cross times q to the power of 5 over denominator q to the power of 10 end fraction   find x.

Answer:

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 to the power of 12 cross times q to the power of 5 equals q to the power of 12 plus 5 end exponent equals q to the power of 17

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 17 over q to the power of 10 equals q to the power of 17 minus 10 end exponent equals q to the power of 7

Write out both sides of the equation

q to the power of x equals q to the power of 7 

Both sides are now over the same base of q

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

x equals 7

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Previous:Collecting Like TermsNext:Expanding & Simplifying Single Brackets
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Jamie Wood

    Reviewer: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.