Let's first simplify the given expression step by step:

The original expression is:

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

Step 1: Simplify each part of the expression

• $(x^6)^{10} = x^{60}$
• The numerator becomes $x^{60} \cdot w^6 \cdot y^6$.
• The denominator remains $w^2 x^2 y^{12}$.

So now the expression looks like:

$\frac{x^{60} \cdot w^6 \cdot y^6}{w^2 \cdot x^2 \cdot y^{12}}$

Step 2: Apply the laws of exponents

• For $x$, $x^{60} \div x^2 = x^{60 - 2} = x^{58}$,
• For $w$, $w^6 \div w^2 = w^{6 - 2} = w^4$,
• For $y$, $y^6 \div y^{12} = y^{6 - 12} = y^{-6}$.

So the simplified expression becomes:

$x^{58} \cdot w^4 \cdot y^{-6}$

This is the final simplified form of the given expression.

Now let's check which of the provided expressions match this simplified form:

1. $\frac{(x^6)^5}{w^3 y^2}$:
   Simplifies to $\frac{x^{30}}{w^3 y^2}$, which is not equivalent.

2. $\frac{x^{30}}{w^3 y^2}$:
   This is already in simplified form, which is not equivalent.

3. $\frac{w^4 x^{58}}{y^6}$:
   Simplifies to $x^{58} \cdot w^4 \cdot y^{-6}$, which is equivalent.

4. $\frac{x^{14}}{y^6} \cdot w^4$:
   Simplifies to $x^{14} \cdot w^4 \cdot y^{-6}$, which is not equivalent.

5. $\left( w^2 x^7 y^3 \right)^2$:
   Simplifies to $w^4 \cdot x^{14} \cdot y^6$, which is not equivalent.

6. $\left( w^2 x^{29} y^3 \right)^2$:
   Simplifies to $w^4 \cdot x^{58} \cdot y^6$, which is not equivalent.

7. $(x^3)^{10} \cdot w^4 y^{-2}$:
   Simplifies to $x^{30} \cdot w^4 \cdot y^{-2}$, which is not equivalent.

Conclusion:

The correct expression that matches the simplified form is $\frac{w^4 x^{58}}{y^6}$.

Would you like further clarification on any step, or details on any rule used?

Related Questions:

1. How do you apply the laws of exponents to simplify expressions like this?
2. What are the common mistakes when dealing with exponent rules?
3. How do negative exponents affect the position of terms in a fraction?
4. Can you explain the process of simplifying a fraction with exponents in more detail?
5. How do you identify equivalent expressions in complex algebraic problems?

Tip: Always simplify both the numerator and denominator separately before applying exponent rules to the entire expression.