Question

prove that
$\sqrt{5 } + \sqrt{3}$
is a rational number.​

Answer 1

If possible suppose √5 + √3 is an rational number .

So, it can be written in the form of p/q where p and q are integers and q ≠ 0.

Then, p/q = √5 + √3

=> p/q = √5 + √3

=> (p/q)²=(√5+√3)² [squaring on both the side]

=> p²/q²= 5 + 3 + 2√15

=> p²/q² = 8 + 2√15

=>p²/q²-8/2 = √15

=> p²- 16q²/2q² = √15

L.H.S. = p²- 16q²/ 2q is a rational number .

Because p and q are co- primes and q≠ 0 and square of a rational number is also a rational number.

therefore, RHS is also a rational number.

But it contradicts the fact that √15 is an irrational number .

And rational no. ≠ irrational number

So, our supposition is wrong.

√5 + √ 3 is an irrational number.