Question

We are given that $a$ when multiplied by $\frac{(\sqrt{2}+3)(1-\sqrt{2})}{3-\sqrt{2}}$ results in a rational number. Which of the following could be a possible value for $a$?

• A. $5\sqrt{2}+1$
• B. $5\sqrt{2}-1$
• C. $5\sqrt{2}+2$
• D. $5\sqrt{2}-2$

Answer 1

The correct option is B $5\sqrt{2}-1$

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

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

$=\frac{(1-2\sqrt{2})(3+\sqrt{2})}{3^2-(\sqrt{2}{)}^2}(\because (a-b\left)\right(a+b)=a^2-b^2)$

$=\frac{3+\sqrt{2}-6\sqrt{2}-2\times 2}{9-2}$
$=\frac{-5\sqrt{2}-1}{7}$
$=-(\frac{5\sqrt{2}+1}{7})$

Given, $a\times -(\frac{5\sqrt{2}+1}{7})\to \text{Rational.}$

To remove the irrational factor we could multiply by $5\sqrt{2}-1.$

i.e., $-(\frac{5\sqrt{2}+1}{7})\times (5\sqrt{2}-1)$

$=-(\frac{(5\sqrt{2}{)}^2-1^2}{7})$

$=-(\frac{25\times 2-1}{7})$

$=-(\frac{50-1}{7})$

$=-\frac{49}{7}$

$=-7$