Question

write the expression in simplest form.
$\frac{5}{3-sqrt{2}} = \frac{square}{square}$

Explanation:

Step1: Rationalize the denominator

Multiply numerator and denominator by the conjugate $3+\sqrt{2}$:
$\frac{5}{3-\sqrt{2}} \times \frac{3+\sqrt{2}}{3+\sqrt{2}} = \frac{5(3+\sqrt{2})}{(3-\sqrt{2})(3+\sqrt{2})}$

Step2: Expand denominator using difference of squares

Apply $(a-b)(a+b)=a^2-b^2$:
$\frac{5(3+\sqrt{2})}{3^2 - (\sqrt{2})^2} = \frac{15+5\sqrt{2}}{9-2}$

Step3: Simplify the denominator

Calculate the denominator value:
$\frac{15+5\sqrt{2}}{7}$

Answer:

$\frac{15+5\sqrt{2}}{7}$