Question

prove √5−2 is irrational​

Answer 1

Answer:

Given: √2+√5

We need to prove√2+√5 is an irrational number.

Proof:

Let us assume that √2+√5 is a rational number.

A rational number can be written in the form of p/q where p,q are integers and q≠0

√2+√5 = p/q

On squaring both sides we get,

(√2+√5)² = (p/q)²

√2²+√5²+2(√5)(√2) = p²/q²

2+5+2√10 = p²/q²

7+2√10 = p²/q²

2√10 = p²/q² – 7

√10 = (p²-7q²)/2q

p,q are integers then (p²-7q²)/2q is a rational number.

Then √10 is also a rational number.

But this contradicts the fact that √10 is an irrational number.

Our assumption is incorrect

√2+√5 is an irrational number.

Hence proved.

Answer 2

Let assume that  $\sqrt{5}-2$ is rational

then,

$\sqrt{5}-2 =\frac{a}{b}$   , where  a and b are co-prime and b $\neq$ 0

$Now,$

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

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

we can see,

a, b, 2 are co-prime

so,

$(\frac{a}{b}+2)$ is rational number but their value is irrational $\sqrt{5}$ which is against our supposition.

so,

$(\frac{a}{b}+2)$ is not rational

$(\frac{a}{b}+2)$ is  an irrational number

hence,  $\sqrt{5}-2$ is an irrational number