https://file.aiquickdraw.com/mathbot/file/1725450453581wntw59qg.jpg

Let's solve the equation given in the image step by step.

### Equation:
\[
\frac{3^7 \times 5^2}{5^4 \times 5 \times 3 \times 3^{-4}} = \frac{3^w}{c^x}
\]

### Simplifying the Left Side:
First, let's simplify the left side of the equation.

1. **Combine powers of 3**:
   \[
   3^7 \text{ (numerator)} \quad \text{and} \quad 3^{-4} \times 3 = 3^{-4+1} = 3^{-3} \text{ (denominator)}
   \]
   So, the power of 3 in the left side will be:
   \[
   3^{7 - (-3)} = 3^{7+3} = 3^{10}
   \]

2. **Combine powers of 5**:
   \[
   5^2 \text{ (numerator)} \quad \text{and} \quad 5^4 \times 5 = 5^{4+1} = 5^5 \text{ (denominator)}
   \]
   So, the power of 5 in the left side will be:
   \[
   5^{2 - 5} = 5^{-3}
   \]

Thus, the simplified left-hand side is:
\[
\frac{3^{10}}{5^{3}}
\]

### Setting up the equation:
Now, equate the simplified expression to the right-hand side:
\[
\frac{3^{10}}{5^{-3}} = \frac{3^w}{c^x}
\]

From this, we can directly compare the exponents:

- For the base 3:
  \[
  10 = w
  \]

- For the base 5:
  \[
  5^{-3} = c^x
  \]
  Here, \( c = 5 \) and \( x = -3 \).

### Final Answer:
- \( c = 5 \)
- \( w = 10 \)
- \( x = -3 \)

Would you like further details or have any questions?

Here are five related questions you might find interesting:
1. How would the solution change if the exponent on \( 3 \) was altered on the right-hand side?
2. What if the equation had a different base, such as 2 instead of 3?
3. How can we solve similar equations where the exponents are unknown?
4. What happens if the equation is not simplified correctly? 
5. How can we apply logarithms to verify the solution?

**Tip:** Always ensure that you simplify both sides of the equation completely before comparing exponents to avoid errors in your calculations.

Find the values of c, w, and x that satisfy the equation: \[\frac{3^7 \times 5^2}{5^4 \times 5 \times 3 \times 3^{-4}} = \frac{3^w}{c^x}\]

Learn how to solve the exponential equation involving exponents and simplification techniques. This problem covers key algebraic concepts such as the laws of exponents and simplifying rational expressions. Suitable for students in Grades 9-12.

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