Let's solve the given expression:

The expression is:

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

Step 1: Apply the power rule

When we raise a fraction to a power, we apply the exponent to both the numerator and denominator. Here, the exponent is $-2$, so we apply it as:

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

Step 2: Simplify the terms

Now simplify each part:

• For the numerator $(a^3 b^{-1})^{-2}$: $(a^3)^{-2} = a^{3 \times (-2)} = a^{-6}$ $(b^{-1})^{-2} = b^{-1 \times (-2)} = b^2$ Thus, the numerator becomes: $a^{-6} b^2$

• For the denominator $(c^{-1} d^4)^{-2}$: $(c^{-1})^{-2} = c^{-1 \times (-2)} = c^2$ $(d^4)^{-2} = d^{4 \times (-2)} = d^{-8}$ Thus, the denominator becomes: $c^2 d^{-8}$

Step 3: Combine the results

We now combine the simplified numerator and denominator: $\frac{a^{-6} b^2}{c^2 d^{-8}} = a^{-6} b^2 c^{-2} d^8$

Step 4: Convert to positive exponents

To express all terms with positive exponents, we move the terms with negative exponents to the denominator: $\frac{b^2 d^8}{a^6 c^2}$

Final Answer:

The simplified expression with all positive exponents is:

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

Would you like a further breakdown or explanation of any part?

Here are five related questions to deepen your understanding:

1. How do negative exponents affect the base in terms of fraction placement?
2. What happens when you apply a negative exponent to both numerator and denominator in a fraction?
3. Can you simplify an expression with multiple exponents applied to variables at once?
4. What is the rule for multiplying powers with the same base?
5. How would you handle an expression with multiple variables and fractional exponents?

Tip: Always simplify negative exponents by moving them to the denominator or numerator, depending on their position.