Prove that √2 + √5 is irrational.
Pls answer
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A rational number can be written in the form of p/q where p,q are integers.
√2+√5 = p/q
Squaring on both sides,
(√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 supposition is false.
√2+√5 is an irrational number.
Hence proved.
(√2+√5)^2=p^2/q^2
2+5+2√10=p^2/q^2
√10=p^2/q^2-7/2
here lhs is irrational but rhs is rational
so.......