Question

prove that 5^1⁄2 is an irrational number.​

Answer 1

Step-by-step explanation:

let √5 be a rational

then it must be in a form p/q where q is not equal to zero (p and q are co-primes)

√5 = p/q

√5 x q = p

Squaring on both sides

5 q^2 = p^2 -----------(1)

p^2 is divisible by 5

so p is divisible by 5

p = 5c

squaring on both sides

p^2 = 25c^2 ------------(2)

put p^2 in equation (1)

5q^2 = 25c^2

q^2 = 5c^2

so q is divisible by 5

thus p and q have common factor 5

So,there is a contradiction in our assumption

We assumed p and q are co-primes but here they have common factor 5

the above statement contradicts our assumption

therefore √5 is irrational