The given expression is:

$\frac{(x^6)^{10}}{w^2 x^2 y^{12}} \cdot w^6 x^6 y^6$

We will simplify this step-by-step.

Step 1: Simplify $(x^6)^{10}$

By applying the power of a power rule $(a^m)^n = a^{m \cdot n}$: $(x^6)^{10} = x^{60}$

Step 2: Simplify the rest of the expression

Now, we can rewrite the expression as: $\frac{x^{60}}{w^2 x^2 y^{12}} \cdot w^6 x^6 y^6$

Step 3: Combine like terms

Let's handle the $x$'s, $w$'s, and $y$'s separately:

• For $x$: $\frac{x^{60}}{x^2} \cdot x^6 = x^{60 - 2 + 6} = x^{64}$
• For $w$: $\frac{w^6}{w^2} = w^{6 - 2} = w^4$
• For $y$: $\frac{y^6}{y^{12}} = y^{6 - 12} = y^{-6}$

Final Expression:

$\frac{x^{64} w^4}{y^6}$

Now we will compare this with the options provided:

1. $\frac{(x^6)^5}{w^3 y^2}$ simplifies to $\frac{x^{30}}{w^3 y^2}$, which does not match.
2. $\frac{x^{30}}{w^3 y^2}$ does not match.
3. $\frac{w^4 x^{58}}{y^6}$ does not match (because $x^{58}$ is incorrect).
4. $\frac{x^{14}}{y^6 w^4}$ does not match.
5. $(w^2 x^7 y^3)^2$ simplifies to $w^4 x^{14} y^6$, which does not match.
6. $(w^2 x^{29} y^3)^2$ simplifies to $w^4 x^{58} y^6$, which does not match.
7. $(x^3)^{10} w^4 y^{-2}$ simplifies to $x^{30} w^4 y^{-2}$, which does not match.

None of the provided options match the correct simplified form of the expression.

Would you like further clarification or details? Here are five follow-up questions:

1. How do you simplify expressions with negative exponents?
2. How does the power of a product rule work in algebra?
3. What is the process for combining exponents when dividing variables?
4. How can you simplify more complex algebraic fractions?
5. What are some common mistakes to watch out for when simplifying powers?

Tip: Always apply exponent rules carefully, ensuring you work on one variable at a time to avoid errors.