Question

Please answer this question ?????

Answer 1

step-by-step

Prove that √3 + √5 is an irrational number .

Let √3 + √5 be a rational number , say r

then √3 + √5 = r

On squaring both sides,

(√3 + √5)2 = r2

3 + 2 √15 + 5 = r2

8 + 2 √15 = r2

2 √15 = r2 - 8

√15 = (r2 - 8) / 2

Now (r2 - 8) / 2 is a rational number and √15 is an irrational number .

1. Since a rational number cannot be equal to an irrational number . Our assumption that √3 + √5 is rational wrong .

Answer 2

Hi friend!!

Let √3+√5 be a rational number.

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

√3+√5 = p/q

√3 = p/q-√5

Squaring on both sides,

(√3)² = (p/q-√5)²

3 = p²/q²+√5²-2(p/q)(√5)

√5×2p/q = p²/q²+5-3

√5 = (p²+2q²)/q² × q/2p

√5 = (p²+2q²)/2pq

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

Then √5 is also a rational number.

But this contradicts the fact that √5 is an irrational number.

So,our supposition is false.

Therefore, √3+√5 is an irrational number