The given expression is:

\[
\frac{6c^{-1}d^{-3}}{15c^2d^{-11}}
\]

To simplify this, we will follow the steps below:

### Step 1: Simplify the coefficients
\[
\frac{6}{15} = \frac{2}{5}
\]

### Step 2: Simplify the powers of \(c\)
For the variable \(c\), we apply the quotient rule for exponents: \(a^m / a^n = a^{m-n}\):

\[
\frac{c^{-1}}{c^2} = c^{-1 - 2} = c^{-3}
\]

### Step 3: Simplify the powers of \(d\)
For the variable \(d\), again applying the quotient rule for exponents:

\[
\frac{d^{-3}}{d^{-11}} = d^{-3 - (-11)} = d^{-3 + 11} = d^8
\]

### Step 4: Combine all the results
Putting everything together:

\[
\frac{6c^{-1}d^{-3}}{15c^2d^{-11}} = \frac{2}{5} \cdot c^{-3} \cdot d^8 = \frac{2d^8}{5c^3}
\]

Thus, the simplified expression is:

\[
\frac{2d^8}{5c^3}
\]

Would you like further clarification on any part of the solution?

Here are 5 related questions:
1. How do you simplify expressions with multiple variables?
2. What are the rules for adding and subtracting exponents?
3. How do negative exponents affect division?
4. Can you explain the power rule for exponents in more detail?
5. What happens when the exponent is zero?

**Tip:** Always simplify the coefficients and variables separately before combining results when simplifying expressions.

Explore how to simplify the expression 6c^-1d^-3/15c^2d^-11 using the quotient rule for exponents and simplification of coefficients. This problem demonstrates key algebraic concepts, such as handling negative exponents and rational expressions. Suitable for students in Grades 8-10.

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