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IBMaths: AI HLDPRevision Notes1. Number & Algebra1.2 Exponentials & Logs1.2.1 Exponents

Exponents (DP IB Maths: AI HL)

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Laws of Indices

What are the laws of indices?

  • Laws of indices (or index laws) allow you to simplify and manipulate expressions involving exponents
    • An exponent is a power that a number (called the base) is raised to
    • Laws of indices can be used when the numbers are written with the same base
  • The index laws 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
    • x to the power of 1 over n end exponent equals blank n-th root of x
    • x to the power of m over n end exponent equals blank n-th root of x to the power of m end root
  • These laws are not in the formula booklet so you must remember them

How are laws of indices used?

  • You will need to be able to carry out multiple calculations with the laws of indices
    • Take your time and apply each law individually
    • Work with numbers first and then with algebra
  • Index laws only work with terms that have the same base, make sure you change the base of the term before using any of the index laws
    • Changing the base means rewriting the number as an exponent with the base you need
    • For example, 9 to the power of 4 equals left parenthesis 3 squared right parenthesis to the power of 4 equals 3 to the power of 2 cross times 4 end exponent equals 3 to the power of 8
    • Using the above can them help with problems like 9 to the power of 4 divided by 3 to the power of 7 equals 3 to the power of 8 divided by 3 to the power of 7 equals 3 to the power of 1 equals 3

Examiner Tip

  • Index laws are rarely a question on their own in the exam but are often needed to help you solve other problems, especially when working with logarithms or polynomials
  • Look out for times when the laws of indices can be applied to help you solve a problem algebraically 

Worked example

Simplify the following equations:

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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    Amber

    Author: Amber

    Expertise: Maths

    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.