Simplify ^5 sqrt symbol(x^6y^5)/^5 sqrt symbol(x^2 y^2)

To simplify the expression ^5√(x^6y^5)/^5√(x^2y^2), we can simplify the numerator and denominator separately, and then divide the two simplifications.

First, let's simplify the numerator, ^5√(x^6y^5). To do this, we'll break down the exponent of each factor into multiple of 5.

For x^6, we can rewrite it as (x^5) * (x^1).
For y^5, we can rewrite it as (y^5) * (y^0).

Now, we can rewrite ^5√(x^6y^5) as ^5√((x^5)*(x^1)*(y^5)*(y^0)).

Applying the properties of radicals, we know that ^5√(a*b) = (^5√a) * (^5√b).

So, our expression becomes (^5√(x^5)) * (^5√(x^1)) * (^5√(y^5)) * (^5√(y^0)).

Simplifying further, we get x * (^5√x) * y * (^5√(y^0)).

However, note that (^5√(y^0)) = 1, since any number raised to the power of 0 is always 1. Therefore, we can simplify this part to just y.

Finally, the numerator simplifies to x * (^5√x) * y.

Now, let's simplify the denominator, ^5√(x^2y^2).

Applying the same logic as before, we can rewrite x^2 as (x^0) * (x^2), and y^2 as (y^0) * (y^2).

So, ^5√(x^2y^2) becomes ^5√((x^0)*(x^2)*(y^0)*(y^2)).

Using the properties of radicals again, we have (^5√(x^0)) * (^5√(x^2)) * (^5√(y^0)) * (^5√(y^2)).

Again, (^5√(x^0)) and (^5√(y^0)) both simplify to 1 since any number raised to the power of 0 is 1.

Therefore, the denominator simplifies to (^5√(x^2)) * (^5√(y^2)).

Now, let's simplify the expression by dividing the numerator and denominator.

Our simplified expression is:
(x * (^5√x) * y) / (^5√(x^2) * ^5√(y^2)).

This is the simplified form of the given expression.