Question

prove that 3 + 2 root 5 is irrational​

Answer 1

Answer:

Step-by-step explanation:

Let us assume √5 to be a rational number.

So, √5 = p/q (where p and q are integers and q is not equal to 0)

√5 = p/q

q√5 = p

Squaring on both sides

5q^2 = p^2

q^2 = p^2/5

If p^2 is divisible by 5 then p is also divisible by 5

Let us assume

c = p/5

p = 5c

p^2 = 25 c^2

Step 2 :

q^2 = (25c^2)/5

q^2 = 5c^2

c^2 = q^2/5

If q^2 is divisible by 5 then q is also divisible by 5.

So p and q becomes co prime integers.

Our assumption is wrong that √5 is a rational number.

So we conclude that √5 is irrational.

Let us assume 3+2√5 to be a rational number

So,

3+2√5 = p/q (where p and q are integers and q is not equal to 0)

3+2√5 = p/q

2√5 = (p – 5q)/q

√5 = (p – 5q)/2q

But we proved that √5 is irrational.

So our assumption is wrong that 3+2√5 is a rational number.

So we conclude that 3+2√5 is a irrational number.

Hope it helps you