Prove that

$fraction numerator 0.75 cross times 0.75 cross times 0.75 plus 0.25 cross times 0.25 cross times 0.25 over denominator 0.75 cross times 0.75 minus 0.75 cross times 0.25 plus 0.25 cross times 0.25 end fraction equals 1$

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Solution

fraction numerator 0.75 cross times 0.75 cross times 0.75 plus 0.25 cross times 0.25 cross times 0.25 over denominator 0.75 cross times 0.75 minus 0.75 cross times 0.25 plus 0.25 cross times 0.25 end fraction equals space fraction numerator left parenthesis 0.75 right parenthesis cubed plus left parenthesis 0.25 right parenthesis cubed over denominator left parenthesis 0.75 right parenthesis squared plus left parenthesis 0.75 right parenthesis left parenthesis 0.25 right parenthesis plus left parenthesis 0.25 right parenthesis squared end fraction equals space fraction numerator left parenthesis 0.75 plus 0.25 right parenthesis left curly bracket left parenthesis 0.75 right parenthesis squared minus left parenthesis 0.75 right parenthesis left parenthesis 0.25 right parenthesis plus left parenthesis 0.25 right parenthesis squared right curly bracket over denominator left parenthesis 0.75 right parenthesis squared left parenthesis 0.75 right parenthesis left parenthesis 0.25 right parenthesis plus left parenthesis 0.25 right parenthesis squared end fraction equals space 1

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