Question

prove that root2- 3 root 5 is irrational

Answer 1

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

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...