https://file.aiquickdraw.com/mathbot/file/1729909927403x6shnsjy.jpg

The problem in the image asks to simplify the expression using the quotient rule:

\[
\left( \frac{z^5}{27x^6} \right)^{-3}
\]

### Step-by-Step Solution:

1. **Apply the power of a quotient rule**:
   \[
   \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}
   \]
   Here, \(a = z^5\), \(b = 27x^6\), and \(n = -3\). So, we rewrite the expression as:
   \[
   \frac{(z^5)^{-3}}{(27x^6)^{-3}}
   \]

2. **Simplify each component**:
   - For the numerator: \((z^5)^{-3} = z^{-15}\)
   - For the denominator: \((27x^6)^{-3} = 27^{-3} \cdot x^{-18}\)

3. **Rewrite the expression**:
   \[
   \frac{z^{-15}}{27^{-3} \cdot x^{-18}}
   \]

4. **Apply the negative exponents rule** (\(a^{-n} = \frac{1}{a^n}\)):
   \[
   z^{-15} = \frac{1}{z^{15}}, \quad 27^{-3} = \frac{1}{27^3}, \quad x^{-18} = \frac{1}{x^{18}}
   \]

   Thus, the expression becomes:
   \[
   \frac{\frac{1}{z^{15}}}{\frac{1}{27^3 \cdot x^{18}}} = \frac{27^3 \cdot x^{18}}{z^{15}}
   \]

5. **Calculate \(27^3\)**:
   \[
   27^3 = (3^3)^3 = 3^9 = 19683
   \]

6. **Final simplified expression**:
   \[
   \frac{19683 \cdot x^{18}}{z^{15}}
   \]

The simplified form is:

\[
\frac{19683x^{18}}{z^{15}}
\]

Would you like further details or explanations?

### Related Questions:
1. How does the quotient rule work with fractional exponents?
2. Can you explain the power of a power rule with an example?
3. How do you handle negative exponents in expressions?
4. What happens when you raise a number to a fractional power?
5. How can you simplify expressions with mixed exponents?

### Tip:
Remember, a negative exponent indicates a reciprocal. For example, \(a^{-n} = \frac{1}{a^n}\).

Use the quotient rule to simplify (z^5 / 27x^6)^-3.

This page provides a step-by-step solution to simplifying (z^5 / 27x^6)^-3 using the quotient rule and negative exponent rule. Ideal for students in Grades 9-11, the solution covers key concepts in exponents.

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