Question

1. when $4 + \sqrt{2}$ is divided by its conjugate, the result is

(1) $\frac{9 + 4\sqrt{2}}{7}$
(2) $-1$
(3) $\frac{11 - 3\sqrt{2}}{6}$
(4) $\frac{4 + 3\sqrt{2}}{4}$

Explanation:

Step1: Find the conjugate of \(4 + \sqrt{2}\)

The conjugate of a binomial \(a + b\sqrt{c}\) is \(a - b\sqrt{c}\), so the conjugate of \(4 + \sqrt{2}\) is \(4 - \sqrt{2}\).

Step2: Set up the division

We need to compute \(\frac{4 + \sqrt{2}}{4 - \sqrt{2}}\). To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is \(4 + \sqrt{2}\).
\[
\frac{(4 + \sqrt{2})(4 + \sqrt{2})}{(4 - \sqrt{2})(4 + \sqrt{2})}
\]

Step3: Expand the numerator and denominator

• Numerator: \((4 + \sqrt{2})(4 + \sqrt{2}) = 4^2 + 2\times4\times\sqrt{2}+(\sqrt{2})^2 = 16 + 8\sqrt{2}+ 2 = 18 + 8\sqrt{2}\)
• Denominator: Using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), we have \((4 - \sqrt{2})(4 + \sqrt{2}) = 4^2-(\sqrt{2})^2 = 16 - 2 = 14\)

Step4: Simplify the fraction

\[
\frac{18 + 8\sqrt{2}}{14}=\frac{2(9 + 4\sqrt{2})}{14}=\frac{9 + 4\sqrt{2}}{7}
\]

Answer:

(1) \(\frac{9 + 4\sqrt{2}}{7}\)