Question

Prove that 5root7 is irrational​

Answer 1

Answer:

See Explanation

Step-by-step explanation:

Let us assume that $5\sqrt 7$ is a rational.

Therefore, it can be expressed in the form of $\frac{p}{q}$, where p and q are integers.

$\implies 5\sqrt 7=\frac{p}{q}$

$\implies \sqrt 7=\frac{p}{5q}$

$\because$ p and q are integers.

$\therefore \frac{p}{5q}$ is a rational number.

$\implies \sqrt 7$ is rational ($\because$ quotient of a rational number is rational)

But, it contradicts the fact that $\sqrt 7$is irrational.

So, $5\sqrt 7$ is also irrational, because the product of rational and irrational is irrational.

This contradiction is arising out because of our wrong assumption that $5\sqrt 7$ is rational.

Hence, $5\sqrt 7$ is irrational.