https://file.aiquickdraw.com/mathbot/file/17270029883279sg6itcd.jpg

The expression provided in the image is:

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

We are asked to simplify this expression to obtain its positive exponents. Let's proceed step by step:

### Step 1: Apply the outer exponent \((-2)\)
We can apply the power rule for exponents \(\left(\frac{X}{Y}\right)^n = \frac{X^n}{Y^n}\), which gives us:

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

### Step 2: Apply exponent \(-2\) to each term inside
Now, we apply \((-2)\) to each of the factors inside both the numerator and the denominator:

- \((a^3)^{-2} = a^{3 \times -2} = a^{-6}\)
- \((b^{-1})^{-2} = b^{-1 \times -2} = b^2\)
- \((c^{-1})^{-2} = c^{-1 \times -2} = c^2\)
- \((d^4)^{-2} = d^{4 \times -2} = d^{-8}\)

### Step 3: Write the new expression
The expression now becomes:

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

### Step 4: Simplify to make all exponents positive
To make all exponents positive, move terms with negative exponents from numerator to denominator (or vice versa):

- \(a^{-6}\) goes to the denominator as \(a^6\),
- \(d^{-8}\) moves to the numerator as \(d^8\).

This gives us the simplified expression:

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

### Final Answer:
The expression with all positive exponents is:

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

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:
1. What is the general rule for handling negative exponents?
2. How do we simplify expressions with multiple negative exponents?
3. Can you explain how the power of a quotient works in terms of exponents?
4. How would the simplification change if the outer exponent were positive instead of negative?
5. How can we apply these rules to more complex algebraic fractions?

**Tip**: Always apply exponents step by step, starting from the innermost operations.

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Learn how to simplify the expression (a^3 b^-1 / c^-1 d^4)^-2 to obtain all positive exponents. This algebra problem involves applying the rules for powers of quotients and negative exponents. Suitable for high school students in Grades 9-12.

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