https://file.aiquickdraw.com/mathbot/file/1727002739802ebe2s66a.jpg

Let's analyze the mathematical expression provided in the image and simplify it.

The given expression is:

\[
\left( \frac{a^3 b^{-1}}{c^{-1} d^4} \right)^{-2}
\]

### Step-by-Step Simplification:

1. **Simplify the exponents inside the parentheses**:
   The expression inside the parentheses is \(\frac{a^3 b^{-1}}{c^{-1} d^4}\).

   - The negative exponent of \(b\) means \(b^{-1} = \frac{1}{b}\).
   - The negative exponent of \(c\) means \(c^{-1} = \frac{1}{c}\).

   This transforms the expression to:

   \[
   \frac{a^3 \cdot \frac{1}{b}}{\frac{1}{c} \cdot d^4} = \frac{a^3}{b} \cdot \frac{c}{d^4} = \frac{a^3 c}{b d^4}
   \]

2. **Apply the exponent -2 to the whole expression**:
   Now, we need to apply the exponent of \(-2\) to the entire fraction \(\frac{a^3 c}{b d^4}\).

   Using the power rule \(\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}\), we apply the exponent \(-2\) to both the numerator and denominator:

   \[
   \left( \frac{a^3 c}{b d^4} \right)^{-2} = \frac{\left( a^3 c \right)^{-2}}{\left( b d^4 \right)^{-2}}
   \]

   Simplifying both the numerator and denominator:

   - \((a^3 c)^{-2} = a^{-6} c^{-2}\)
   - \((b d^4)^{-2} = b^{-2} d^{-8}\)

   So the expression becomes:

   \[
   \frac{a^{-6} c^{-2}}{b^{-2} d^{-8}} = a^{-6} c^{-2} \cdot \frac{1}{b^{-2} d^{-8}} = a^{-6} c^{-2} b^2 d^8
   \]

3. **Convert to positive exponents**:
   Now, to express everything with positive exponents, we get:

   \[
   b^2 d^8 a^{-6} c^{-2} = \frac{b^2 d^8}{a^6 c^2}
   \]

Thus, the simplified expression with all positive exponents is:

\[
\boxed{\frac{b^2 d^8}{a^6 c^2}}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do you apply negative exponents to a fraction?
2. What is the general rule for simplifying an expression with multiple terms raised to a negative exponent?
3. How does changing the exponent sign affect the position of variables in the numerator or denominator?
4. How do you simplify expressions with both positive and negative exponents?
5. How does the power of a product rule work when applying exponents?

**Tip:** When simplifying expressions with exponents, always manage the signs carefully and apply exponent rules step by step to avoid mistakes.

Pangkat positif dari \(\left( \frac{a^3 b^{-1}}{c^{-1} d^4} \right)^{-2}\) adalah ....

This problem involves simplifying an algebraic expression with multiple variables and negative exponents. Learn how to apply the rules of exponents to simplify \(\left( \frac{a^3 b^{-1}}{c^{-1} d^4} \right)^{-2}\) and express it with positive exponents.

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