Mathematics

Problem

Consider the general expression

\displaystyle \frac{1}{a+b\sqrt{x}}

, where

a, b, x

are positive integers and

x

is not a perfect square. a. (i) Write the conjugate of

a+b\sqrt{x}

. b. (ii) Prove that

(a+b\sqrt{x})(a-b\sqrt{x})

is a rational number. c. (iii) Using

a=2

b=1

and

x=3

, rationalize the denominator of

\displaystyle \frac{1}{2+\sqrt{3}}

and show explicitly that the new denominator is rational.