The problem asks to: a) Express $\frac{\sqrt{3}+1}{\sqrt{3}-1} + \sqrt{3} - 1$ in the form $a + b\sqrt{3}$ where $a$ and $b$ are rational numbers. b) Rationalize the denominator of each of the following expressions: (i) $\frac{2\sqrt{3}-\sqrt{2}}{4\sqrt{3}}$ (ii) $\frac{x}{x+\sqrt{y}}$ (iii) $\frac{2\sqrt{7}+\sqrt{3}}{3\sqrt{7}-\sqrt{3}}$ (iv) $\frac{x-\sqrt{x^2-9}}{x+\sqrt{x^2-9}}$ (v) $\frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)}$ c) Rationalize the numerator of each of the following expressions: (i) $\frac{\sqrt{5+h}-3}{h}$ (ii) $\frac{\sqrt{3}+\sqrt{5}}{7}$ (iii) $\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}+\sqrt{x+h}}$

1. Problem Description

The problem asks to:

a) Express $\frac{\sqrt{3}+1}{\sqrt{3}-1} + \sqrt{3} - 1$ in the form $a + b\sqrt{3}$ where $a$ and $b$ are rational numbers.

b) Rationalize the denominator of each of the following expressions:

(i) $\frac{2\sqrt{3}-\sqrt{2}}{4\sqrt{3}}$

(ii) $\frac{x}{x+\sqrt{y}}$

(iii) $\frac{2\sqrt{7}+\sqrt{3}}{3\sqrt{7}-\sqrt{3}}$

(iv) $\frac{x-\sqrt{x^2-9}}{x+\sqrt{x^2-9}}$

(v) $\frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)}$

c) Rationalize the numerator of each of the following expressions:

(i) $\frac{\sqrt{5+h}-3}{h}$

(ii) $\frac{\sqrt{3}+\sqrt{5}}{7}$

(iii) $\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}+\sqrt{x+h}}$

2. Solution Steps

a) Express $\frac{\sqrt{3}+1}{\sqrt{3}-1} + \sqrt{3} - 1$ in the form $a + b\sqrt{3}$:

First, we rationalize the denominator of the first term:

$\frac{\sqrt{3}+1}{\sqrt{3}-1} = \frac{(\sqrt{3}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = \frac{3 + 2\sqrt{3} + 1}{3-1} = \frac{4+2\sqrt{3}}{2} = 2+\sqrt{3}$.

Then, we add the remaining terms:

$2 + \sqrt{3} + \sqrt{3} - 1 = 1 + 2\sqrt{3}$.

Thus, $a=1$ and $b=2$.

b) Rationalize the denominator of each expression:

(i) $\frac{2\sqrt{3}-\sqrt{2}}{4\sqrt{3}} = \frac{(2\sqrt{3}-\sqrt{2})\sqrt{3}}{4\sqrt{3}\sqrt{3}} = \frac{6 - \sqrt{6}}{12}$

(ii) $\frac{x}{x+\sqrt{y}} = \frac{x(x-\sqrt{y})}{(x+\sqrt{y})(x-\sqrt{y})} = \frac{x(x-\sqrt{y})}{x^2-y} = \frac{x^2 - x\sqrt{y}}{x^2-y}$

(iii) $\frac{2\sqrt{7}+\sqrt{3}}{3\sqrt{7}-\sqrt{3}} = \frac{(2\sqrt{7}+\sqrt{3})(3\sqrt{7}+\sqrt{3})}{(3\sqrt{7}-\sqrt{3})(3\sqrt{7}+\sqrt{3})} = \frac{6(7) + 2\sqrt{21} + 3\sqrt{21} + 3}{9(7)-3} = \frac{42 + 5\sqrt{21} + 3}{63-3} = \frac{45+5\sqrt{21}}{60} = \frac{9+\sqrt{21}}{12}$

(iv) $\frac{x-\sqrt{x^2-9}}{x+\sqrt{x^2-9}} = \frac{(x-\sqrt{x^2-9})(x-\sqrt{x^2-9})}{(x+\sqrt{x^2-9})(x-\sqrt{x^2-9})} = \frac{x^2 - 2x\sqrt{x^2-9} + x^2 - 9}{x^2 - (x^2-9)} = \frac{2x^2-9 - 2x\sqrt{x^2-9}}{9}$

(v) $\frac{1}{(\sqrt{2}+1)(\sqrt{3}-1)} = \frac{(\sqrt{2}-1)(\sqrt{3}+1)}{(\sqrt{2}+1)(\sqrt{2}-1)(\sqrt{3}-1)(\sqrt{3}+1)} = \frac{\sqrt{6}+\sqrt{2}-\sqrt{3}-1}{(2-1)(3-1)} = \frac{\sqrt{6}+\sqrt{2}-\sqrt{3}-1}{2}$

c) Rationalize the numerator of each expression:

(i) $\frac{\sqrt{5+h}-3}{h} = \frac{(\sqrt{5+h}-3)(\sqrt{5+h}+3)}{h(\sqrt{5+h}+3)} = \frac{(5+h)-9}{h(\sqrt{5+h}+3)} = \frac{h-4}{h(\sqrt{5+h}+3)}$

(ii) $\frac{\sqrt{3}+\sqrt{5}}{7} = \frac{(\sqrt{3}+\sqrt{5})(\sqrt{3}-\sqrt{5})}{7(\sqrt{3}-\sqrt{5})} = \frac{3-5}{7(\sqrt{3}-\sqrt{5})} = \frac{-2}{7(\sqrt{3}-\sqrt{5})} = \frac{2}{7(\sqrt{5}-\sqrt{3})}$

(iii) $\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}+\sqrt{x+h}} = \frac{(\sqrt{x}-\sqrt{x+h})(\sqrt{x}+\sqrt{x+h})}{(h\sqrt{x}+\sqrt{x+h})(\sqrt{x}+\sqrt{x+h})} = \frac{x-(x+h)}{(h\sqrt{x}+\sqrt{x+h})(\sqrt{x}+\sqrt{x+h})} = \frac{-h}{(h\sqrt{x}+\sqrt{x+h})(\sqrt{x}+\sqrt{x+h})} = \frac{-h}{h x + h\sqrt{x}\sqrt{x+h} + \sqrt{x}\sqrt{x+h} + x + h} = \frac{-h}{h x + x + h + (h+1)\sqrt{x(x+h)}} = \frac{-1}{x + (x+h)/h+ (1+(1/h))\sqrt{x^2+xh}}$

3. Final Answer

a) $1 + 2\sqrt{3}$

b)

(i) $\frac{6 - \sqrt{6}}{12}$

(ii) $\frac{x^2 - x\sqrt{y}}{x^2-y}$

(iii) $\frac{9+\sqrt{21}}{12}$

(iv) $\frac{2x^2-9 - 2x\sqrt{x^2-9}}{9}$

(v) $\frac{\sqrt{6}+\sqrt{2}-\sqrt{3}-1}{2}$

c)

(i) $\frac{h-4}{h(\sqrt{5+h}+3)}$

(ii) $\frac{2}{7(\sqrt{5}-\sqrt{3})}$

(iii) $\frac{-1}{hx+x+h + (h+1)\sqrt{x(x+h)}}$