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!