Question

remove the radical from denominator
select one:

simplify and rationalize the denominator
select one:

Explanation:

for First Problem:

Step1: Rationalize denominator

Multiply numerator and denominator by $\sqrt{2}$: $\frac{2 \times \sqrt{2}}{3\sqrt{2} \times \sqrt{2}}$

Step2: Simplify denominator

$\sqrt{2} \times \sqrt{2} = 2$, so denominator is $3 \times 2 = 6$

Step3: Simplify fraction

$\frac{2\sqrt{2}}{6} = \frac{\sqrt{2}}{3}$ (Note: Assuming a typo in options, closest valid logic aligns with simplification; if original numerator was 4, it would be $\frac{4\sqrt{2}}{6} = \frac{2\sqrt{2}}{3}$, but given options, likely intended numerator adjustment leads to $\frac{4 - 2\sqrt{2}}{7}$ being incorrect, but based on standard rationalization, the process is as above.)

for Second Problem:

Step1: Rationalize denominator

Multiply numerator and denominator by $\sqrt{3} + \sqrt{2}$: $\frac{\sqrt{8}(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}$

Step2: Simplify denominator

$(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1$

Step3: Simplify numerator

$\sqrt{8} = 2\sqrt{2}$, so $2\sqrt{2}(\sqrt{3} + \sqrt{2}) = 2\sqrt{6} + 4$; none of the options match, so choose "None".

Answer:

First problem: (Assuming intended correction, but based on options, no exact match; if forced, but standard rationalization gives $\frac{\sqrt{2}}{3}$)
Second problem: B. None