Question
Prove that following numbers are irrationals:
$5\sqrt{2}$
$5\sqrt{2}$
✓
Answer
Let us assume that $5\sqrt{2}$ is rational.
Then, there exist positive co-primes a and b such that,
$5\sqrt{2}=\frac{\text{a}}{\text{b}}$
$\sqrt{2}=\frac{\text{a}}{5\text{b}}$
$\sqrt{2}$ is a rational number which is a contradication.
Hence $5\sqrt{2}$ is irrational.
Then, there exist positive co-primes a and b such that,
$5\sqrt{2}=\frac{\text{a}}{\text{b}}$
$\sqrt{2}=\frac{\text{a}}{5\text{b}}$
$\sqrt{2}$ is a rational number which is a contradication.
Hence $5\sqrt{2}$ is irrational.
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