Let's simplify the expression step by step using exponent rules. The given expression is:

$\left( \frac{6y^{-3}}{12y^{-5}} \right)^3$

Step 1: Simplify the fraction inside the parentheses

First, simplify the numerical part and the variables separately:

• Numerical Part: $\frac{6}{12} = \frac{1}{2}$
• Variable Part: Use the rule $\frac{a^m}{a^n} = a^{m-n}$ for the $y$ terms. So: $\frac{y^{-3}}{y^{-5}} = y^{-3 - (-5)} = y^{-3 + 5} = y^2$

Now the expression becomes:

$\left( \frac{1}{2} y^2 \right)^3$

Step 2: Apply the power of 3 to both terms

Now distribute the exponent 3 to both the fraction and the variable $y^2$:

• Numerical Part: $\left( \frac{1}{2} \right)^3 = \frac{1^3}{2^3} = \frac{1}{8}$
• Variable Part: $(y^2)^3 = y^{2 \times 3} = y^6$

Thus, the entire expression simplifies to:

$\frac{y^6}{8}$

Final Answer:

The simplified expression is $\frac{y^6}{8}$, which corresponds to the last option in the multiple choice answers.

Would you like more details or have any other questions?

Here are 5 related questions to expand on this concept:

1. How do you simplify expressions with negative exponents?
2. What is the general rule for raising a fraction to a power?
3. Can you explain how the exponent rules apply to multiplication and division of variables?
4. How do you simplify expressions with multiple variables and exponents?
5. What happens if there are zero exponents in an expression?

Tip: Always remember to combine like terms before applying exponent rules to simplify expressions efficiently.