Question
Classify the following number as rational or irrational:$\big(5+3\sqrt{2}\big)$
✓
Answer
Let $5+3\sqrt2$ be rational.
Hence, $$5 and $5+3\sqrt2$ are rational.
$\therefore\big(5+3\sqrt{2}-2\big)=3\sqrt2=$ rational $[\because$ Difference of two rational is rational$]$
$\therefore\frac{1}3{}\times3\sqrt{2}=\sqrt{2}=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt2$ is irrational.
The contradiction arises by assuming $5+3\sqrt2$ is rational.
Hence, $5+3\sqrt2$ is irrational.
Hence, $$5 and $5+3\sqrt2$ are rational.
$\therefore\big(5+3\sqrt{2}-2\big)=3\sqrt2=$ rational $[\because$ Difference of two rational is rational$]$
$\therefore\frac{1}3{}\times3\sqrt{2}=\sqrt{2}=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt2$ is irrational.
The contradiction arises by assuming $5+3\sqrt2$ is rational.
Hence, $5+3\sqrt2$ is irrational.
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