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Real Numbers — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsReal Numbers2 Marks
Question
Given that $\sqrt{2}$ is irrational, prove that $(5+3 \sqrt{2})$ is an irrational number.
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Answer

Suppose $(5+3 \sqrt{2})=\frac{p}{q}$Now assume $(5+3 \sqrt{2})$ is a rational number.
Therefore $p$ and $q$ should be co$-$prime numbers.
$(5+3 \sqrt{2})=\frac{p}{q}$
$\Rightarrow \frac{p}{q}-5=3 \sqrt{2}$
$\Rightarrow \frac{p}{3 q}-\frac{5}{3}=\sqrt{2}$
$\Rightarrow \frac{p-5}{3 q}=\sqrt{2}$
Since $\sqrt{2}$ is irrational number.
Thus the assumption is incorrect and hence $(5+3 \sqrt{2})$ is an irrational number.
Hence proved.

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