Simplify. $\newline$ $\frac{2}{3+\sqrt{2}}$

Full solution

Q. Simplify.

\newline

\frac{2}{3+\sqrt{2}}

Rationalize Denominator: Rationalize the denominator of the expression $\frac{2}{3+\sqrt{2}}$ . To remove the square root from the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $3+\sqrt{2}$ is $3-\sqrt{2}$ . $\frac{2}{3+\sqrt{2}} \cdot \frac{3-\sqrt{2}}{3-\sqrt{2}}$
Apply Distributive Property: Apply the distributive property (also known as the FOIL method) to the denominator. $\newline$ $(3+\sqrt{2})(3-\sqrt{2}) = 3^2 - (\sqrt{2})^2$
Calculate Squares: Calculate the squares in the denominator. $3^2 - (\sqrt{2})^2 = 9 - 2$
Simplify Denominator: Simplify the denominator. $9 - 2 = 7$
Apply Distributive Property: Apply the distributive property to the numerator. $\newline$ $2(3) - 2(\sqrt{2}) = 6 - 2\sqrt{2}$
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified form of the expression is $(6 - 2\sqrt{2})/7$ .