Question
Classify the following number as rational or irrational:$\big(2-3\sqrt5\big)$
✓
Answer
Let $2-3\sqrt5$ be rational.
Hence 2 and $2-3\sqrt5$ are rational.
$\therefore2-\big(2-3\sqrt5\big)=2-2+3\sqrt5$
$=3\sqrt5=$ rational $[\because$ Difference of two rational is rational$]$
$\therefore\frac{1}{3}\times3\sqrt5=\sqrt5=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt5$ is irrational.
The contradiction arises by assuming $2-3\sqrt5$ is rational.
Hence, $2-3\sqrt5$ is irrational.
Hence 2 and $2-3\sqrt5$ are rational.
$\therefore2-\big(2-3\sqrt5\big)=2-2+3\sqrt5$
$=3\sqrt5=$ rational $[\because$ Difference of two rational is rational$]$
$\therefore\frac{1}{3}\times3\sqrt5=\sqrt5=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt5$ is irrational.
The contradiction arises by assuming $2-3\sqrt5$ is rational.
Hence, $2-3\sqrt5$ is irrational.
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