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Real Numbers — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsReal Numbers1 Mark
Question
Classify the following number as rational or irrational:$\frac{3}{\sqrt{5}}$
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Answer

Let $\frac{3}{\sqrt5}$ be rational.
$\therefore\frac{1}3{}\times\frac{3}{\sqrt5}=\frac{1}{\sqrt5}=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\frac{1}{\sqrt5}$ is irrational.
$\therefore\frac{1\times\sqrt5}{\sqrt5\times\sqrt5}=\frac{1}5{}\sqrt5$
So, if $\frac{1}{\sqrt5}$ is rational, then $\frac{1}{5}\sqrt5$ is rational.
$\therefore5\times\frac{1}{5}\sqrt5=\sqrt5=$ rational $[\because$ Product of two rational is rational$]$
Hence, $\frac{1}{\sqrt5}$ is irrational.
The contradiction arises by assuming $\frac{3}{\sqrt5}$ is rational.
Hence, $\frac{3}{\sqrt5}$ is irrational.

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