Question

$prove \: \sqrt{2} + \sqrt{5 \: } \:is \: irrational$
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Answer 1

Step-by-step explanation:

Let √2+√5 be a rational number.

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.

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Answer 2

Let √2+√5 be a rational number.

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.

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