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IBMathsDPApplications & Interpretation (AI)SLRevision Notes1. Number & AlgebraNumber ToolkitLaws of Indices

Laws of Indices (DP IB Applications & Interpretation (AI)): Revision Note

Amber

Written by: Amber

Reviewed by: Mark Curtis

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DP Applications & Interpretation (AI): SLDP Applications & Interpretation (AI): SLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

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Laws of Indices

What are the index laws?

  • The index laws that you need to know are:

    • left parenthesis x y right parenthesis to the power of m equals x to the power of m end exponent y to the power of m

    • stretchy left parenthesis x over y stretchy right parenthesis to the power of m equals x to the power of m over y to the power of m

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

    • x to the power of m divided by x to the power of n equals x to the power of m minus n end exponent

    • stretchy left parenthesis x to the power of m stretchy right parenthesis to the power of n equals x to the power of m n end exponent

    • x to the power of 1 equals x

    • x to the power of 0 equals 1

    • 1 over x to the power of m equals x to the power of negative m end exponent

Examiner Tips and Tricks

The index laws are not in the formula booklet so you must remember them!

Examiner Tips and Tricks

For AI SL, you do not need to know about fractional indices (for HL you do).

How do I change the base?

  • Index laws only work with terms that have the same base

    • 2 cubed cross times 5 squared cannot be simplified using index laws

  • You can sometimes rewrite a base as a power of another base

    • 2 to the power of 5 cross times bold 4 cubed equals 2 to the power of 5 cross times open parentheses bold 2 to the power of bold 2 close parentheses cubed

      • The 4 changes to open parentheses 2 squared close parentheses

      • This is called changing the base

    • It can now be simplified using index laws

    • 2 to the power of 5 cross times open parentheses 2 squared close parentheses cubed equals 2 to the power of 5 cross times 2 to the power of 6 equals 2 to the power of 11

Worked Example

Simplify the following expressions:

i) fraction numerator left parenthesis 3 x squared right parenthesis left parenthesis 2 x cubed y squared right parenthesis over denominator left parenthesis 6 x squared y right parenthesis end fraction

 

ai-sl-1-1-2-laws-of-indices-we-i

ii) left parenthesis 4 x squared y to the power of negative 4 end exponent right parenthesis cubed left parenthesis 2 x cubed y to the power of negative 1 end exponent right parenthesis to the power of negative 2 end exponent

 

ai-sl-1-1-2-laws-of-indices-we-ii


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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    Mark Curtis

    Reviewer: 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.