Question

show that root 5 and root 7 are irrational

Answer 1

We will prove whether √7 is irrational by contradiction method. Let √7 be rational It can be expressed as √7 = a/b ( where a, b are integers and co-primes.
√7=a/b

7= a²/b² 
7b² = a²
7divides a²
By the Fundamental theorem of Arithmetic 
so, 7 divides a .

a = 7k (for some integer) 

a² = 49k² 
7b² = 49k² 
b² = 7k² 

7divides b²
7 divides b. 

Now 7 divides both a & b this contradicts the fact that they are co primes. 
this happened due to faulty assumption that √7 is rational. Hence, √7 is irrational. 


We will prove whether √5 is irrational by contradiction method.
Let √5 be rational 
It can be expressed as √5 = a/b ( where a, b are integers and co-primes. 
√5 = a/b
5= a²/b² 
5b² = a²
5 divides a²
By the Fundamental theorem of Arithmetic
so, 5 divides a .

a = 5k (for some integer) 

a² = 25k² 
5b² = 25k² 
b² = 5k² 

5 divides b²
5 divides b. 

Now 5 divides both a & b this contradicts the fact that they are co primes. 
this happened due to faulty assumption that √5 is rational. Hence, √5 is irrational.