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Polynomials

Question

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Simplify the following into a fraction with rational denominator $fraction numerator 1 over denominator square root of 5 plus square root of 6 minus 11 end fraction.$

Solution

fraction numerator 1 over denominator square root of 5 plus square root of 6 minus square root of 11 end fraction equals space fraction numerator 1 over denominator square root of 5 plus square root of 6 minus square root of 11 end fraction space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator square root of 5 plus square root of 6 plus square root of 11 end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator left parenthesis square root of 5 plus square root of 6 right parenthesis squared minus left parenthesis square root of 11 right parenthesis squared end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator 5 plus 6 plus 2 square root of 30 minus 11 end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator 2 square root of 30 end fraction equals space fraction numerator square root of 5 plus square root of 6 plus square root of 11 over denominator 2 square root of 30 end fraction fraction numerator square root of 30 over denominator square root of 30 end fraction equals space fraction numerator square root of 150 plus square root of 180 plus square root of 330 over denominator 60 end fraction equals space fraction numerator 5 square root of 6 plus 6 square root of 5 plus square root of 330 over denominator 60 end fraction

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