Mathematics

Problem

Consider the binomial expression

a+b\sqrt{c}

, where

a

and

b

are rational numbers and

c

is a positive integer that is not a perfect square. a. (i) Define the term "conjugate" of the binomial

a+b\sqrt{c}

. Describe its form in one sentence. b. (ii) Prove that the product of a binomial containing a square root and its conjugate is always a rational number by showing that

(a+b\sqrt{c})(a-b\sqrt{c})=a^{2}-b^{2}c

. c. (iii) Using the result in part (ii), explain why the denominator becomes rational when the fraction

\dfrac{1}{4+\sqrt{7}}

is multiplied by

4-\sqrt{7}

both in the numerator and the denominator.

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