Question

Prove that 6+✓2 is a irrational number​

Answer 1

Prove that $\sf{6+\sqrt{2}}$

Step-by-step explanation:

Assume that $\sf{6+\sqrt{2}}$ is rational.

So, $\sf{6+\sqrt{2}=\dfrac{a}{b}\:---(a\:and\:b\:are\:integers)}$

$\longrightarrow{\sf{\sqrt{2}=\dfrac{a}{b} - 6}}$

$\longrightarrow{\sf{\sqrt{2}=\dfrac{a - 6b}{b}}}$

$\sf{\dfrac{a - 6b}{b}}$ is a rational number as a and b are integers.

So, it means that $\sf{\sqrt{2}}$ is rational. But this a contradiction to the fact that $\sf{\sqrt{2}}$ is irrational.

The contradiction was arisen due to wrong assumption.

Hence proved, that $\sf{6+\sqrt{2}}$ is irrational.