The number 5+2 5-2 is - Vidyadip

MCQ

The number $\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$ is

✓
an irrational number
B
an integer
C
not a real number
D
a rational number

✓

Answer

Correct option: A.

an irrational number

$=\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$
$=\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}} \times \frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}}$
$=\frac{(\sqrt{5}+\sqrt{2})^2}{(\sqrt{5})^2-(\sqrt{2})^2}$
$=\frac{(\sqrt{5})^2+(\sqrt{2})^2+2 \times \sqrt{5} \times \sqrt{2}}{5-2}$
$=\frac{5+2+2 \sqrt{10}}{3}$
$=\frac{7+2 \sqrt{10}}{3}$
Here $\sqrt{10}=\sqrt{2} \times \sqrt{5}$
Since $\sqrt{2}$ and $\sqrt{5}$ both are an irrational number
Therefore, $\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$ is an irrational number.

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