Question

prove that root 5 + 3 root 2 is irrational

Answer 1

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Answer 2

Let √5 + 3√2 be a rational number.

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

√5 + 3√2 = p/q

√5 = p/q - 3√2

squaring on both sides,

√5² = (p/q - 3√2)²

5 = p²/q² - 2(p/q)(3√2) + (3√2)²

5 = p²/q² - 6√2p/q + 18

p²/q² + 13 = 6√2p/q

(p² + 13q²)/q² = 6√2p/q

(p²+13q²)/q² × q/6p = √2

(p²+13q²)/6pq = √2

p,q are integers then (p²+13q²)/6pq must be an integer

then √2 must be an integer.

But this contradicts the face that √2 is an irrational number.

Therefore,our supposition is false.

√5 + 3√2 is an irrational number.

hence proved.

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