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Polynomials

Question

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Rationalise the denominators of the following:

$fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction$

Solution

fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction equals fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction cross times fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction

Multiplying the numerator and denominator by

square root of 3 minus 1

equals fraction numerator left parenthesis square root of 3 minus 1 right parenthesis squared over denominator open parentheses square root of 3 close parentheses squared minus left parenthesis 1 right parenthesis squared end fraction equals fraction numerator left parenthesis square root of 3 right parenthesis squared minus 2 left parenthesis square root of 3 right parenthesis left parenthesis 1 right parenthesis plus left parenthesis 1 right parenthesis squared over denominator 3 minus 1 end fraction equals space fraction numerator 3 minus 2 square root of 3 plus 1 over denominator 2 end fraction equals fraction numerator 4 minus 2 square root of 3 over denominator 2 end fraction equals 2 minus square root of 3.

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