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Algebraic Roots & Indices (CIE IGCSE Maths: Extended)

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Algebraic Roots & Indices

Can I use the laws of indices with algebra?

  • Laws of indices work with numerical and algebraic terms
  • These can be used to simplify expressions where terms are multiplied or divided
    • Deal with the number and algebraic 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 11
      • fraction numerator 3 x to the power of 7 over denominator 6 x to the power of 4 end fraction equals 3 over 6 cross times x to the power of 7 over x to the power of 4 equals 1 half x cubed
      • 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
  • The index laws you need to know and use are summarised here:List of index laws, A Level & AS Level Pure Maths Revision Notes

How can I solve equations when the unknown is in the index?

  • If two powers (bigger than 1) are equal and the base numbers are the same then the indices must be the same
    • If a to the power of x equals a to the power of y then  x equals y
  • If the unknown is part of the index then write both sides with the same base number
    • Then you can ignore the base number and make the indices equal and solve that equation

table row cell 5 to the power of 2 x end exponent end cell equals 125 row cell 5 to the power of 2 x end exponent end cell equals cell 5 cubed end cell row cell 2 x space end cell equals cell space 3 end cell row cell x space end cell equals cell space 3 over 2 end cell end table

  • In more complicated questions you might have to use negative and fractional indices
    • You may also have to rewrite both sides with the same base number

table attributes columnalign right center left columnspacing 0px end attributes row cell 8 to the power of x end cell equals cell 1 fourth end cell row cell open parentheses 2 cubed close parentheses to the power of x end cell equals cell 1 over 2 squared end cell row cell 2 to the power of 3 x end exponent end cell equals cell 2 to the power of negative 2 end exponent end cell row cell 3 x end cell equals cell negative 2 end cell row x equals cell negative 2 over 3 end cell end table

Worked example

(a)table row cell blank to the power of blank end cell row blank end table
Simplify  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.


Multiply out the brackets in the numerator.

Rearrange the numerator so that you are multiplying the numbers together, the x terms together and the y terms together.

fraction numerator 3 cross times 2 cross times x squared cross times x cubed cross times y squared over denominator 6 x squared y end fraction

Simplify the numerator.
Multiply the constants together and add the powers of the x terms together.

fraction numerator 6 x to the power of 5 y squared over denominator 6 x squared y end fraction

Divide the constants.
Subtract the power of the x term in the denominator from the x term in the numerator: x to the power of 5 minus 2 end exponent equals x cubed.
Subtract the power of the y term in the denominator from the y term in the numerator: y to the power of 2 minus 1 end exponent equals y to the power of 1.

bold italic x to the power of bold 3 bold italic y

(b)table row blank row blank end table
Simplify  open parentheses fraction numerator 54 x to the power of 7 over denominator 2 x to the power of 4 end fraction close parentheses to the power of negative 1 third end exponent.

Simplify the expression inside the brackets:
Cancel down the constants (divide 54 by 2).
Subtract the power of the x term in the denominator from the x term in the numerator: x to the power of 7 minus 4 end exponent equals x cubed.

open parentheses 27 x cubed close parentheses to the power of negative 1 third end exponent

Apply the negative index outside the brackets by 'flipping' the fraction inside the brackets.

open parentheses fraction numerator 1 over denominator 27 x cubed end fraction close parentheses to the power of 1 third end exponent

Apply the fractional index outside the brackets to everything inside the brackets.

fraction numerator 1 to the power of 1 third end exponent over denominator 27 to the power of 1 third end exponent x to the power of 3 cross times 1 third end exponent end fraction

Simplify.

fraction numerator bold 1 over denominator bold 3 bold italic x end fraction 

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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.