By what number should `(8//27)^(-3)` be a divided to that the quotient is equal to `(27//8)^(-3)` ?

Explanation

To solve the problem, we need to find the number by which \((\frac{8}{27})^{-3}\) should be divided so that the quotient equals \((\frac{27}{8})^{-3}\).

### Step-by-Step Solution:

1. **Write the equation based on the problem statement:**
We need to find a number \( x \) such that:
\[ \frac{(\frac{8}{27})^{-3}}{x} = (\frac{27}{8})^{-3} \]

2. **Rearranging the equation:**
To isolate \( x \), we can multiply both sides by \( x \):
\[ (\frac{8}{27})^{-3} = x \cdot (\frac{27}{8})^{-3} \]

3. **Using the property of negative exponents:**
Recall that \( a^{-m} = \frac{1}{a^m} \). Therefore:
\[ (\frac{27}{8})^{-3} = \frac{1}{(\frac{27}{8})^3} = \frac{8^3}{27^3} \]
So we can rewrite the equation as:
\[ (\frac{8}{27})^{-3} = x \cdot \frac{8^3}{27^3} \]

4. **Calculating \((\frac{8}{27})^{-3}\):**
Again using the property of negative exponents:
\[ (\frac{8}{27})^{-3} = \frac{27^3}{8^3} \]

5. **Substituting back into the equation:**
Now we have:
\[ \frac{27^3}{8^3} = x \cdot \frac{8^3}{27^3} \]

6. **Cross-multiplying to solve for \( x \):**
Multiply both sides by \( 27^3 \) and \( 8^3 \):
\[ 27^3 \cdot 27^3 = x \cdot (8^3 \cdot 8^3) \]
This simplifies to:
\[ 27^6 = x \cdot 8^6 \]

7. **Isolating \( x \):**
Divide both sides by \( 8^6 \):
\[ x = \frac{27^6}{8^6} \]

8. **Simplifying the expression:**
This can be rewritten as:
\[ x = \left(\frac{27}{8}\right)^6 \]

### Final Answer:
Thus, the number by which \((\frac{8}{27})^{-3}\) should be divided is:
\[ \left(\frac{27}{8}\right)^6 \]