Question

1. $\frac{\sqrt{3}}{-1 - \sqrt{5}}$ 10) $\frac{\sqrt{5}}{5 + \sqrt{2}}$ 11) $\frac{2 - \sqrt{3}}{-2 - \sqrt{5}}$ 12) $\frac{-4 + \sqrt{3}}{-1 - 2\sqrt{5}}$

Explanation:

Problem 9: Rationalize $\boldsymbol{\frac{\sqrt{3}}{-1 - \sqrt{5}}}$

Step 1: Identify the conjugate

The denominator is $-1 - \sqrt{5}$, so its conjugate is $-1 + \sqrt{5}$. Multiply numerator and denominator by the conjugate.
$$\frac{\sqrt{3}}{-1 - \sqrt{5}} \times \frac{-1 + \sqrt{5}}{-1 + \sqrt{5}}$$

Step 2: Multiply numerators and denominators

Numerator: $\sqrt{3}(-1 + \sqrt{5}) = -\sqrt{3} + \sqrt{15}$
Denominator: $(-1)^2 - (\sqrt{5})^2 = 1 - 5 = -4$

Step 3: Simplify the fraction

$$\frac{-\sqrt{3} + \sqrt{15}}{-4} = \frac{\sqrt{3} - \sqrt{15}}{4}$$

Step 1: Identify the conjugate

The denominator is $5 + \sqrt{2}$, so its conjugate is $5 - \sqrt{2}$. Multiply numerator and denominator by the conjugate.
$$\frac{\sqrt{5}}{5 + \sqrt{2}} \times \frac{5 - \sqrt{2}}{5 - \sqrt{2}}$$

Step 2: Multiply numerators and denominators

Numerator: $\sqrt{5}(5 - \sqrt{2}) = 5\sqrt{5} - \sqrt{10}$
Denominator: $5^2 - (\sqrt{2})^2 = 25 - 2 = 23$

Step 1: Identify the conjugate

The denominator is $-2 - \sqrt{5}$, so its conjugate is $-2 + \sqrt{5}$. Multiply numerator and denominator by the conjugate.
$$\frac{2 - \sqrt{3}}{-2 - \sqrt{5}} \times \frac{-2 + \sqrt{5}}{-2 + \sqrt{5}}$$

Step 2: Multiply numerators and denominators

Numerator: $(2 - \sqrt{3})(-2 + \sqrt{5}) = -4 + 2\sqrt{5} + 2\sqrt{3} - \sqrt{15}$
Denominator: $(-2)^2 - (\sqrt{5})^2 = 4 - 5 = -1$

Step 3: Simplify the fraction

$$\frac{-4 + 2\sqrt{5} + 2\sqrt{3} - \sqrt{15}}{-1} = 4 - 2\sqrt{5} - 2\sqrt{3} + \sqrt{15}$$

Answer:

$\boldsymbol{\frac{\sqrt{3} - \sqrt{15}}{4}}$

Problem 10: Rationalize $\boldsymbol{\frac{\sqrt{5}}{5 + \sqrt{2}}}$