Introduction:
In this question, we have to prove that 5√7 is an irrational number. An irrational number is a number that cannot be expressed as a ratio of two integers. In other words, it cannot be written in the form of p/q, where p and q are integers and q ≠ 0.

Assumption:
Let us assume that 5√7 is a rational number. This means that we can write 5√7 in the form of p/q, where p and q are integers and q ≠ 0. We can also assume that p and q have no common factors.

Squaring Both Sides:
Now, let's square both sides of the equation 5√7 = p/q.

(5√7)² = (p/q)²

Simplifying, we get:

175 = p²/q²

Conclusion:
Now, we can see that p² is divisible by 7, which means that p is also divisible by 7. Let's write p as 7x, where x is an integer.

Substituting this value of p in the equation, we get:

175 = (7x)²/q²

Simplifying, we get:

25 = x²/q²

Now, we can see that x² is divisible by 5, which means that x is also divisible by 5. But this contradicts our assumption that p and q have no common factors. Therefore, our assumption that 5√7 is a rational number is incorrect, and 5√7 is an irrational number.