Question

Prove that 2 + 5 √3 is an irrational number​

Answer 1

Question

Prove that $2+5\sqrt{3}$ is an irrational Number

Answer

Let us assume that $2+5\sqrt{3}$ is a rational number. Then,

$\implies$ $2+5\sqrt{3}$=$\large\frac{a}{b}$

$\implies$ $5\sqrt{3}=\large\frac{a}{b}-2$

$\implies$ $5\sqrt{3}=\large\frac{a-2b}{b}$

$\implies$ $\sqrt{3}=\large\frac{a-2b}{5b}$

Therefore $\frac{a-2b}{5b}$ is in form of $\frac{a}{b}$ which is in rational number.

But, this contradicts the fact that $\sqrt{3}$ is an irrational Number.

Therefore our assumption is wrong and $2+5\sqrt{3}$ is an irrational number.