If $fraction numerator 2 over denominator square root of 3 plus square root of 5 end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction equals straight a square root of 3 plus straight b square root of 5 comma$ find a and b.

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Solution

fraction numerator 2 over denominator square root of 3 plus square root of 5 end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction equals a square root of 3 plus b square root of 5 rightwards double arrow space space space fraction numerator 2 left parenthesis square root of 3 minus square root of 5 right parenthesis over denominator left parenthesis square root of 3 plus square root of 5 right parenthesis left parenthesis square root of 3 minus square root of 5 right parenthesis end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction space fraction numerator square root of 3 plus square root of 5 over denominator square root of 3 plus square root of 5 end fraction space equals a square root of 3 plus b square root of 5 rightwards double arrow space space space space fraction numerator 2 left parenthesis square root of 3 minus square root of 5 right parenthesis over denominator 3 minus 5 end fraction plus fraction numerator 5 left parenthesis square root of 3 plus square root of 5 right parenthesis over denominator 3 minus 5 end fraction space equals space a square root of 3 plus b square root of 5 rightwards double arrow space space minus left parenthesis square root of 3 minus square root of 5 right parenthesis minus 5 over 2 left parenthesis square root of 3 plus square root of 5 right parenthesis space equals space a square root of 3 plus b square root of 5 equals negative 7 over 2 square root of 3 minus 3 over 2 square root of 5 equals a square root of 3 plus b square root of 5 rightwards double arrow space a equals negative 7 over 2. space space space space space space b equals negative 3 over 2 space

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Polynomials

If $fraction numerator 2 over denominator square root of 3 plus square root of 5 end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction equals straight a square root of 3 plus straight b square root of 5 comma$ find a and b.

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