Let's simplify the given expression step by step:

Given Expression:

$\frac{y^6}{x^{-1}x^{-5}y^{-8}}$

Step 1: Combine the terms in the denominator.

In the denominator, we have $x^{-1}x^{-5}$. Using the rule of exponents $(a^m \cdot a^n = a^{m+n})$, this simplifies to: $x^{-1}x^{-5} = x^{-6}.$

So, the expression becomes: $\frac{y^6}{x^{-6}y^{-8}}.$

Step 2: Use the division rule of exponents.

The rule is $\frac{a^m}{a^n} = a^{m-n}$.

• For $x$: The numerator has no $x$, so it simplifies to $x^{0 - (-6)} = x^6$.
• For $y$: $\frac{y^6}{y^{-8}} = y^{6 - (-8)} = y^{6+8} = y^{14}$.

Thus, the expression simplifies to: $x^6y^{14}.$

Final Answer:

The equivalent expression is: $x^6y^{14}.$

This matches the fourth option in the choices provided.

Would you like more explanation or examples? Here are five related questions:

1. How do you simplify expressions with negative exponents in general?
2. What happens if there are variables in the numerator and denominator with the same base?
3. How do you simplify expressions like $(x^{-2})(x^3)/(x^{-4})$?
4. Can you explain why $a^0 = 1$ for any $a \neq 0$?
5. What’s the difference between adding exponents and multiplying them?

Tip: Always combine terms with the same base first before applying division or multiplication rules.