Question

question 13 of 25
how would the fraction \\(\frac{1}{3+\sqrt{2}}\\) be rewritten if its denominator is rationalized using difference of squares?

a. \\(\frac{3-\sqrt{2}}{5}\\)

b. \\(3 - \sqrt{2}\\)

c. \\(\frac{3-\sqrt{2}}{7}\\)

d. \\(-\frac{3+\sqrt{2}}{7}\\)

Explanation:

Step1: Identify conjugate of denominator

Conjugate of $3+\sqrt{2}$ is $3-\sqrt{2}$.

Step2: Multiply numerator and denominator by conjugate

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

Step3: Simplify denominator using difference of squares

$$(3)^2 - (\sqrt{2})^2 = 9 - 2 = 7$$

Step4: Write simplified fraction

$$\frac{3-\sqrt{2}}{7}$$

Answer:

C. $\frac{3-\sqrt{2}}{7}$