Answer
$\dfrac{\sqrt{14}-2\sqrt 2}{3}$
Work Step by Step
Rationalize the denominator by multiplying $\sqrt7-2$ to both the numerator and the denominator:
$\dfrac{\sqrt 2}{\sqrt 7+2}\\
=\dfrac{\sqrt 2}{\sqrt 7+2} \cdot \dfrac{\sqrt 7-2}{\sqrt 7-2}$
Use special formula $(a+b)(a-b)=a^2-b^2$ in the denominator.
$=\dfrac{\sqrt 2(\sqrt 7-2)}{(\sqrt 7)^2-2^2}$
$=\dfrac{\sqrt 2(\sqrt 7-2)}{7-4}$
$=\dfrac{\sqrt 2(\sqrt 7-2)}{3}$
Apply the distributive property.
$=\dfrac{\sqrt2\cdot \sqrt 7-\sqrt2(2)}{3}$
$=\dfrac{\sqrt2\cdot \sqrt 7-2\sqrt2}{3}$
Use the rule $\sqrt{a} \cdot \sqrt{b}=\sqrt{a\cdot b}$ then simplify:
$=\dfrac{\sqrt{2\cdot 7}-2\sqrt2}{3}$
$=\dfrac{\sqrt{14}-2\sqrt2}{3}$
Hence, the correct answer is $\frac{\sqrt{14}-2\sqrt2}{3}$.
