Maths Question

Suppose that 5³ˡ ² = √5³ = 5√5 is a rational number.
So, we have 5√5 = p/q ,
√5 = p/(5q) = P/Q where P and Q are relatively prime integers.
Hence, we have 5 = P²/Q² ,
5Q² = P²
Then, P² is divisible by 5 and hence P is divisible by 5.
Therefore, P = 5h for some integer h.
Hence, we have 5Q² = (5h)²
Q² = 5h²
So, Q² is divisible by 5 and hence Q is divisible by 5.
It leads to a contradiction since P and Q are relatively prime.

Thus, we have 5³ˡ ² is an irrational number.