can you guide me through the steps to simplify this problem: (9^(-z+1/7)(1/243)^((-4/9y)-(2/7))(1/243)^(5/2))/(27^(1/3x)(27)^((-5/8y)-(3/7))(9)^((2x+2/7))

Certainly! Let’s simplify the expression step by step. The given expression is:

\[
\frac{9^{(-z + \frac{1}{7})} \cdot (1/243)^{(-\frac{4}{9}y - \frac{2}{7})} \cdot (1/243)^{\frac{5}{2}}}{27^{\frac{1}{3}x} \cdot 27^{(-\frac{5}{8}y - \frac{3}{7})} \cdot 9^{(2x + \frac{2}{7})}}
\]

### Step 1: Convert Bases

First, express all numbers in terms of powers of 3:

- \( 9 = 3^2 \)
- \( 27 = 3^3 \)
- \( 243 = 3^5 \)

Now we rewrite the expression using these conversions:

1. \( 9^{(-z + \frac{1}{7})} = (3^2)^{(-z + \frac{1}{7})} = 3^{2(-z + \frac{1}{7})} = 3^{-2z + \frac{2}{7}} \)
2. \( (1/243) = (3^{-5}) \), so:
   \[
   (1/243)^{(-\frac{4}{9}y - \frac{2}{7})} = (3^{-5})^{(-\frac{4}{9}y - \frac{2}{7})} = 3^{5(\frac{4}{9}y + \frac{2}{7})}
   \]
   \[
   (1/243)^{\frac{5}{2}} = (3^{-5})^{\frac{5}{2}} = 3^{- \frac{25}{2}}
   \]
3. \( 27^{\frac{1}{3}x} = (3^3)^{\frac{1}{3}x} = 3^{x} \)
4. \( 27^{(-\frac{5}{8}y - \frac{3}{7})} = (3^3)^{(-\frac{5}{8}y - \frac{3}{7})} = 3^{(-3)(\frac{5}{8}y + \frac{3}{7})} = 3^{- \frac{15}{8}y - \frac{9}{7}} \)
5. \( 9^{(2x + \frac{2}{7})} = (3^2)^{(2x + \frac{2}{7})} = 3^{2(2x + \frac{2}{7})} = 3^{4x + \frac{4}{7}} \)

### Step 2: Substitute Back into the Expression

Now substitute these values back into the original expression:

\[
\frac{3^{-2z + \frac{2}{7}} \cdot 3^{\frac{20}{9}y + \frac{10}{7}} \cdot 3^{- \frac{25}{2}}}{3^{x} \cdot 3^{- \frac{15}{8}y - \frac{9}{7}} \cdot 3^{4x + \frac{4}{7}}}
\]

### Step 3: Combine the Exponents

Combine the exponents in the numerator and denominator:

**Numerator:**
\[
-2z + \frac{2}{7} + \frac{20}{9}y + \frac{10}{7} - \frac{25}{2}
\]

**Denominator:**
\[
x - \left(\frac{15}{8}y + \frac{9}{7}\right) + (4x + \frac{4}{7})
\]

### Step 4: Simplify Each Part

1. **Numerator:**
   Combine the constant terms and coefficients of \(y\):
   \[
   -2z + \left(\frac{20}{9} - \frac{15}{8}\right)y + \left(\frac{2}{7} + \frac{10}{7} - \frac{25}{2}\right)
   \]

   Calculate \( \frac{20}{9} - \frac{15}{8} \) and \( \frac{2 + 10 - 87.5}{7} = \frac{-75.5}{7} \).

2. **Denominator:**
   Combine the terms:
   \[
   (x + 4x) - \frac{15}{8}y - \frac{9}{7} + \frac{4}{7} = 5x - \frac{15}{8}y - \frac{5}{7}
   \]

### Step 5: Final Result

After combining and simplifying, write the final expression as:

\[
3^{\text{(numerator exponent - denominator exponent)}}
\]

Would you like to see the calculations for the exact values of the exponents, or do you have any questions about the steps? 

### Additional Questions:
1. What is the significance of converting numbers to a common base?
2. How do you determine the common denominator for adding fractions?
3. Can you provide an example of simplifying a different expression using these steps?
4. How would you approach similar problems with different bases?
5. What are some common pitfalls to avoid in these types of simplifications?

### Tip:
Always keep track of negative exponents and ensure you simplify fractions correctly to avoid errors in your calculations!

Can you guide me through the steps to simplify this problem: (9^(-z+1/7)(1/243)^((-4/9y)-(2/7))(1/243)^(5/2))/(27^(1/3x)(27)^((-5/8y)-(3/7))(9)^((2x+2/7))

Discover how to simplify complex expressions with exponents step-by-step. This guide includes detailed explanations and calculations for the expression involving bases of 9, 27, and 243, suitable for high school students.

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