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Algebraic Roots & Indices

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) begin mathsize 16px style table row cell blank to the power of blank end cell row blank end table end style
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.
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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