Question

prove that √2-3√5 is an irrational number
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Answer 1

Answer:

Let √2-3√5 is a rational number

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

√2-3√5=p/q

Squaring on both sides,

(√2-3√5)²=(p/q)²

[√2²+(3√5)²-2(√2)(3√5)]=p²/q²

[2+9(5)-6√10]=p²/q²

[2+45-6√10]=p²/q²

6√10=p²/q²-47

6√10=(p²-47q²)/q²

√10=(p²-47q²)6q²

p,q are integers then (p²-47q²)/6q² is a rational number.

Then,√10 is also a rational number.

But this contradicts the fact that √10 is a rational number.

So,our supposition is false.

Hence,√2-3√5 is irrational number.

Hope it helps

Answer 2

nice question

Let  2-3√5 be a rational no.

therefore 2-3√5=a/b

thus √5=a-2b/3b

therefore a-2b/3b is a rational no.

therefore √5 is a rational no

but we know that √5 is a irrational no

therefore our assumption was wrong that 2-3√5 is a rational no

therefore 2-3√5 is a irrational number