Answer:
See Explanation
Step-by-step explanation:
Let us assume that [tex] 5\sqrt 7[/tex] is a rational.
Therefore, it can be expressed in the form of [tex] \frac{p}{q}[/tex], where p and q are integers.
[tex] \implies 5\sqrt 7=\frac{p}{q}[/tex]
[tex] \implies \sqrt 7=\frac{p}{5q}[/tex]
[tex] \because [/tex] p and q are integers.
[tex] \therefore \frac{p}{5q}[/tex] is a rational number.
[tex] \implies \sqrt 7[/tex] is rational ([tex] \because [/tex] quotient of a rational number is rational)
But, it contradicts the fact that [tex]\sqrt 7[/tex]is irrational.
So, [tex]5\sqrt 7[/tex] is also irrational, because the product of rational and irrational is irrational.
This contradiction is arising out because of our wrong assumption that [tex] 5\sqrt 7[/tex] is rational.
Hence, [tex] 5\sqrt 7[/tex] is irrational.
