Question

. Prove that 5-2 root3 is an irrational number, given that root3 is an irrational numbe​

Answer 1

Answer:

Ajmer know that root 3 is irrational number.

then the sum of the rational and irrational will be rational number.

similarly, 5-2 root 3 will be irrational number.

Answer 2

GIVEN :

• $\sqrt {3}$ is an irrational number.

TO PROVE :

• $\sf 5 \ - \ 2 \sqrt {3} \ is \ an \ irrational \ number$

SOLUTION :

To prove $\sf 5 \ - \ 2 \sqrt {3}$ is an irrational number, Assume it is rational number.

Rational number are written in the form of $\sf \dfrac {a}{b}$. where a and b are integers.

$\implies \sf 5 \ - \ 2 \sqrt {3} \ = \ \dfrac {a}{b}$

$\implies \sf -2 \sqrt {3} \ = \ - \dfrac {a}{b}$

$\implies \sf 2 \sqrt {3} \ = \ 5 \ - \ ab$

$\implies \sf 2 \sqrt {3} \ = \ \dfrac {5b}{b} \ - \ \dfrac {a}{b}$

$\implies \sf \sqrt {3} \ = \ 5b \ - \ \dfrac {a}{2b}$

Now,

Here we know that the numbers 2, 5,. a and b are integers and rational number.

$\therefore$ Our assumption that it is an rational number is false.

Therefore, $\sf 5 \ - \ 2 \sqrt {3}$ is an irrational number.

•°• Hence proved ✔︎