The problem is to simplify the expression $\frac{6x^3y^5}{2x^6y^2}$.

1. Problem Description

The problem is to simplify the expression $\frac{6x^3y^5}{2x^6y^2}$.

2. Solution Steps

We can simplify the given expression by dividing the coefficients and applying the quotient rule for exponents.

The quotient rule for exponents is:

$\frac{a^m}{a^n} = a^{m-n}$

First, divide the coefficients:

$\frac{6}{2} = 3$

Next, apply the quotient rule to the x terms:

$\frac{x^3}{x^6} = x^{3-6} = x^{-3}$

Then, apply the quotient rule to the y terms:

$\frac{y^5}{y^2} = y^{5-2} = y^3$

Combining these results gives us:

$3x^{-3}y^3$

We can rewrite $x^{-3}$ as $\frac{1}{x^3}$, so the simplified expression is:

$3\frac{1}{x^3}y^3 = \frac{3y^3}{x^3}$

3. Final Answer

$\frac{3y^3}{x^3}$