Prove that $fraction numerator 2 to the power of 30 plus 2 to the power of 29 plus 2 to the power of 28 over denominator 2 to the power of 31 plus 2 to the power of 30 minus 2 to the power of 29 end fraction equals 7 over 10$

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Solution

fraction numerator 2 to the power of 30 plus 2 to the power of 29 plus 2 to the power of 28 over denominator 2 to the power of 31 plus 2 to the power of 30 minus 2 to the power of 29 end fraction space space space space space space space space space space space equals space fraction numerator 2 to the power of 28 left parenthesis 2 squared plus 2 to the power of 1 plus 1 right parenthesis over denominator 2 to the power of 29 left parenthesis 2 squared plus 2 to the power of 1 minus 1 right parenthesis end fraction space space space space space space space space space space space equals space fraction numerator 2 squared plus 2 to the power of 1 plus 1 over denominator 2 to the power of 29 minus 28 end exponent left parenthesis 2 squared plus 2 to the power of 1 minus 1 right parenthesis end fraction space space space space space space space space space space space equals space space fraction numerator 4 plus 2 plus 1 over denominator 2 to the power of 1 left parenthesis 4 plus 2 plus negative 1 right parenthesis end fraction space space space space space space space space space space space space equals space fraction numerator 7 over denominator 2 left parenthesis 5 right parenthesis end fraction space space space space space space space space space space space space equals space 7 over 10

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