Question

6.
Prove that 5-2√3 is an irrational number.​

Answer 1

Answer:

Given:3 + 2√5

To prove:3 + 2√5 is an irrational number.

Proof:

Letus assume that 3 + 2√5 is a rational number.

Soit can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving3 + 2√5 = a/b we get,

=>2√5 = a/b – 3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/2b

This shows (a-3b)/2b is a rational number. But we know that But √5 is an irrational number.

so it contradictsour assumption.

Our assumption of3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved

Answer 2

Step-by-step explanation:

Let us assume that "5 - 2√3" is a rational number

Where ,

• p, q are co-primes and q ≠ 0

→ 5 - 2√3 = $\frac{p}{q}$

→ -2√3 = $\frac{p}{q}$ - 5

→ -2√3 = $\frac{5q~-~p}{q}$

→ √3 = $\frac{5q~-~p}{q} {+} {2}$

→ √3 = $\frac{5q~-~p}{2q}$

Here, $\frac{5q~-~p}{2q}$ is a rational where p, q are co-primes and q ≠ 0

We know that ,

A rational number cannot be a irrational number

:. Our assumption is wrong

$\huge\red{5-2√3}$ is an irrational.