Question

Which mathematical operator would make the irrational numbers, $\sqrt{5}+\sqrt{6}$ and $\sqrt{6}-\sqrt{5}$, rational?

• A. Addition
• B. Subtraction
• C. Multiplication
• D. Division

Answer 1

The correct option is C

Multiplication

A) Addition: $\sqrt{5}+\sqrt{6}+\sqrt{6}-\sqrt{5}=2\sqrt{6}$, which is an irrational number.

B) Subtraction: $\sqrt{5}+\sqrt{6}-\sqrt{6}+\sqrt{5}=2\sqrt{5}$, which is an irrational number.

C) Multiplication: $\sqrt{5}+\sqrt{6}\times \sqrt{6}-\sqrt{5}={\sqrt{6}}^2-{\sqrt{5}}^2=6-5=1$, which is a rational number.

D) Division: $\frac{\sqrt{5}+\sqrt{6}}{\sqrt{6}-\sqrt{5}}=\frac{\sqrt{5}+\sqrt{6}}{\sqrt{6}-\sqrt{5}}\times \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}$

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

$=11+2\sqrt{30}$, which is an irrational number.