Simplify $\frac{1}{3-√2}$ in the form of p + q√2

Analysis

The goal is to simplify the expression $\frac{1}{3-\sqrt{2}}$ by removing the square root from the denominator. This process is known as rationalizing the denominator.

Concept

To rationalize a denominator in the form $(a - \sqrt{b})$, you multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(3 - \sqrt{2})$ is $(3 + \sqrt{2})$.

Multiplying by the conjugate works because of the difference of two squares identity: $(a - b)(a + b) = a^2 - b^2$

Step-by-Step Solution

Step 1: Multiply the numerator and denominator by the conjugate.

Step 2: Expand the numerator and the denominator. Numerator: $1 \times (3 + \sqrt{2}) = 3 + \sqrt{2}$

Denominator: $(3 - \sqrt{2})(3 + \sqrt{2})$ Using the identity $(a - b)(a + b) = a^2 - b^2$: $= 3^2 - (\sqrt{2})^2$ $= 9 - 2$ $= 7$

Step 3: Combine the results.

Step 4: Express in the form $p + q\sqrt{2}$. Split the fraction:

This matches the form $p + q\sqrt{2}$, where $p = \frac{3}{7}$ and $q = \frac{1}{7}$.

Conclusion

Comparing our result $\frac{3}{7} + \frac{\sqrt{2}}{7}$ with the given options, it matches option D.

Exam Tip: Whenever you see a surd in the denominator, immediately think "conjugate." If the denominator is $(a - \sqrt{b})$, multiply by $(a + \sqrt{b})$. If it is $(a + \sqrt{b})$, multiply by $(a - \sqrt{b})$. This will always clear the root from the bottom!