Prove that 7 5 is irrational. - Vidyadip

Question

Prove that $7 \sqrt { 5 }$ is irrational.

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Answer

We can prove $7 \sqrt { 5 }$ irrational by contradiction.
Let us suppose that $7 \sqrt { 5 }$ is rational.
It means we have some co-prime integers a and b (b≠ 0)

such that
$7 \sqrt { 5 } = \frac { a } { b }$
$\Rightarrow \sqrt { 5 } = \frac { a } { 7 b }$ .......(1)
R.H.S of (1) is rational but we know that $\sqrt { 5 }$ is irrational.
It is not possible which means our supposition is wrong.
Therefore, $7 \sqrt { 5 }$ cannot be rational.
Hence, it is irrational.

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