Question

Prove that $7\surd 5$ Is irrational number.

Answer 1

A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions. Rational numbers are terminating decimals but irrational numbers are non-terminating.

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

Hence, $7\surd 5$can be written in the form of $\frac{a}{b}$ where$a,b$are co-prime and $b$ not equal to $0$.

$7\surd 5=\frac{a}{b}$

$\surd 5=\frac{a}{7b}$

Here, $\surd 5$ is irrational and $\frac{a}{7b}$ is a rational number.

Rational number$\ne$ Irrational number

It is a contradiction to our assumption $7\surd 5$ is a rational number.

Therefore, $7\surd 5$ is an irrational number.

Hence, it is proved that $7\surd 5$ Is irrational number.