Question

Which of the following should be multiplied with $\sqrt{3}+\sqrt{2}$ to get a rational number?

• A. $\sqrt{3}-\sqrt{2}$
• B. $-\sqrt{3}+\sqrt{2}$
• C. $3+\sqrt{2}$
• D. $\sqrt{3}+\sqrt{2}$

Answer 1

Answer: (A), (B)

$-\sqrt{3}+\sqrt{2}$

To get a rational number, we have to multiply the given surds by its conjugate pair.
The conjugate pair of $\sqrt{3}+\sqrt{2}$ can be either $\sqrt{3}-\sqrt{2}$ or $-\sqrt{3}+\sqrt{2}$.

We can check by multiplying the conjugate pair with the given surd as follows
$(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})=(\sqrt{3}{)}^2-(\sqrt{2}{)}^2=3-2=1$

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