Let's solve the given expression from the image:

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

Step 1: Simplify the inner expression

We can simplify the terms inside the parentheses first.

$\frac{3a^2 b c^{-3}}{a b^{-2} c^2}$

1. Simplify the constants:

   • The constant is $3$, so no simplification is needed here.

2. Simplify the powers of $a$:

   • In the numerator, the power of $a$ is $2$.
   • In the denominator, the power of $a$ is $1$.
   • Subtract the exponents: $a^{2-1} = a^1$.

3. Simplify the powers of $b$:

   • In the numerator, the power of $b$ is $1$.
   • In the denominator, the power of $b$ is $-2$.
   • Subtract the exponents: $b^{1 - (-2)} = b^{1 + 2} = b^3$.

4. Simplify the powers of $c$:

   • In the numerator, the power of $c$ is $-3$.
   • In the denominator, the power of $c$ is $2$.
   • Subtract the exponents: $c^{-3 - 2} = c^{-5}$.

Thus, the simplified expression inside the parentheses is:

$3a b^3 c^{-5}$

Step 2: Square the entire expression

Now we square the simplified expression:

$\left( 3a b^3 c^{-5} \right)^2$

1. Square the constant: $3^2 = 9$.

2. Square the powers of $a$: $a^1 \rightarrow (a^1)^2 = a^2$.

3. Square the powers of $b$: $b^3 \rightarrow (b^3)^2 = b^6$.

4. Square the powers of $c$: $c^{-5} \rightarrow (c^{-5})^2 = c^{-10}$.

Thus, the final expression is:

$9a^2 b^6 c^{-10}$

Final Answers:

• Power of $a$: $2$
• Power of $b$: $6$
• Power of $c$: $-10$, but since the answer requires positive powers, we rewrite this as $\frac{1}{c^{10}}$.
• Constant value: $9$
• Positions:
  • $a$ is in the numerator.
  • $b$ is in the numerator.
  • $c$ is in the denominator (because of the negative power).
  • Constant value is in the numerator.

Let me know if you want further clarifications or details. Here are five related questions to consider:

1. How do you apply the power rule for exponents when simplifying fractions?
2. What happens when you raise a fraction to a power?
3. How do you convert negative exponents into fractions?
4. Can you simplify a similar expression where variables are raised to fractional powers?
5. How would the solution change if the original expression involved cube roots?

Tip:

When working with negative exponents, remember that they indicate reciprocals (e.g., $x^{-n} = \frac{1}{x^n}$).