Question

prove that root 7+2root5 is an irrational number

Answer 1

Let √7 + 2√5 is a rational number.
A rational number can be written in the form of p/q where p,q are integers.

√7 + 2√5 = p/q
√7 = p/q - 2√5

Squaring on both sides,
√7² = [p/q - 2√5]²
7 = p²/q² + 4(5) - 2(p/q)(2√5)
7 = p²/q² + 20 - 4√5p/q
p²/q² - 4√5p/q = 7 - 20
p²/q² + 13 = 4√5p/q
(p²+13q²)/q² = 4√5p/q
(p²+13q²)/q² × q/4p = √5
(p²+13q²)/4pq = √5

p,q are integers then (p²+13q²)/4pq 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.
Hence,√7 + 2√5 is an irrational number