Show that the following numbers are irrational. 75 - Vidyadip

Question

Show that the following numbers are irrational.
$7\sqrt{5}$

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Answer

Let us assume that $7\sqrt{5}$ is rational.
Then , there exist positive co primes a and b such that,
$7\sqrt{5}=\frac{\text{a}}{\text{b}}$
$\sqrt{5}=\frac{\text{a}}{7\text{b}}$
We know that $\sqrt{5}$ is an irrational number.
Here we see that $\sqrt{5}$ is a rational number which is a contradiction.

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