Relations and Functions

Relations and Functions

Question

Simplify

space space space space space space space space space space space space fraction numerator left parenthesis 1 space minus straight i right parenthesis cubed over denominator 1 minus straight i cubed end fraction equals fraction numerator left parenthesis 1 minus straight i right parenthesis cubed over denominator straight i cubed minus straight i cubed end fraction

Answer
space space space space space space space space space space space space space fraction numerator left parenthesis 1 minus straight i right parenthesis cubed over denominator 1 minus straight i cubed end fraction equals fraction numerator left parenthesis 1 minus straight i right parenthesis cubed over denominator 1 cubed minus straight i cubed end fraction equals fraction numerator left parenthesis straight i minus straight i right parenthesis cubed over denominator left parenthesis 1 minus straight i right parenthesis left parenthesis 1 plus straight i plus straight i squared right parenthesis end fraction  [ a3 - b3 = (a - b) (a2 + ab +b2)

  space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 1 minus straight i right parenthesis squared over denominator 1 space plus space straight i space plus straight i squared end fraction equals fraction numerator 1 space plus space straight i squared minus 2 straight i over denominator 1 space plus space straight i squared plus straight i end fraction equals fraction numerator 1 minus 1 minus 2 straight i over denominator 1 minus 1 plus straight i end fraction

                space space space space space space space space space space space equals fraction numerator negative 2 straight i over denominator straight i end fraction equals negative 2

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