Question

State whether the following statement is true or false.

The following number is irrational
$6+\sqrt{2}$

• A. True
• B. False

Answer 1

The correct option is A True

An Irrational Number is a real number that cannot be written as a simple fraction.

Let us assume $6+\sqrt{2}$ is rational.

Hence, $6+\sqrt{2}$ can be written in form $\frac{a}{b}$

Where, $a$ and $b$ $(n\ne 0)$ are co-prime.

Hence, $6+\sqrt{2}=\frac{a}{b}$

$\Rightarrow$ $\sqrt{2}=\frac{a}{b}-6$

$\Rightarrow$ $\sqrt{2}=\frac{a-6b}{b}$

Here, $\frac{a-6b}{b}$ is a rational number, but $\sqrt{2}$ is irrational.

Since, Rational $\ne$ Irrational

This is a contriadition

$\therefore$ Our assumption is incorrect.

$\therefore$ $6+\sqrt{2}$ is irrational number.