The image shows three math problems that involve simplifying expressions and evaluating values. I'll break down each one:

Problem 1:

$\left( \frac{a^{-2}b^3c^{-3}}{ab^{-2}c^3} \right)^{-2}$ We need to simplify the expression inside the parentheses, then apply the exponent of $-2$.

Problem 2:

$\left( \frac{x^3y^{-2}z^4}{x^{-2}y^3c^3} \right)^{-1}$ This is another simplification problem, where we simplify the fraction inside the parentheses, followed by applying the exponent of $-1$.

Problem 3:

This involves evaluating the value of the following: $\left( \frac{1}{3} \right)^{-3} \cdot 3^2 \cdot 2^3$ Divided by: $\left( \frac{1}{3} \right)^3 \cdot 3^2 \cdot 2^{-1}$

I will solve each problem step by step.

Problem 1 Solution:

First, simplify the terms inside the parentheses: $\frac{a^{-2}b^3c^{-3}}{ab^{-2}c^3}$

• Combine the exponents of $a$, $b$, and $c$ individually:
  • For $a$: $a^{-2} \div a = a^{-2-1} = a^{-3}$
  • For $b$: $b^3 \div b^{-2} = b^{3-(-2)} = b^{5}$
  • For $c$: $c^{-3} \div c^3 = c^{-3-3} = c^{-6}$

So the simplified expression inside the parentheses is: $\frac{a^{-3}b^5c^{-6}}{1}$ Now apply the exponent $-2$: $\left( a^{-3}b^5c^{-6} \right)^{-2} = a^{6}b^{-10}c^{12}$ Thus, the simplified expression is: $a^6b^{-10}c^{12}$

Problem 2 Solution:

First, simplify the terms inside the parentheses: $\frac{x^3y^{-2}z^4}{x^{-2}y^3c^3}$

• For $x$: $x^3 \div x^{-2} = x^{3-(-2)} = x^{5}$
• For $y$: $y^{-2} \div y^3 = y^{-2-3} = y^{-5}$
• For $z$: $z^4$ (no change as there is no $z$ in the denominator)
• For $c$: $c^{-3}$

So the simplified expression inside the parentheses is: $x^5y^{-5}z^4c^{-3}$ Now apply the exponent $-1$: $\left( x^5y^{-5}z^4c^{-3} \right)^{-1} = x^{-5}y^{5}z^{-4}c^{3}$ Thus, the simplified expression is: $x^{-5}y^5z^{-4}c^3$

Problem 3 Solution:

We need to calculate: $\frac{\left( \frac{1}{3} \right)^{-3} \cdot 3^2 \cdot 2^3}{\left( \frac{1}{3} \right)^3 \cdot 3^2 \cdot 2^{-1}}$

First, simplify the numerator: $\left( \frac{1}{3} \right)^{-3} = 3^3$ $3^3 \cdot 3^2 \cdot 2^3 = 3^{3+2} \cdot 2^3 = 3^5 \cdot 2^3$

Now, simplify the denominator: $\left( \frac{1}{3} \right)^3 = \frac{1}{3^3} = 3^{-3}$ $3^{-3} \cdot 3^2 \cdot 2^{-1} = 3^{-3+2} \cdot 2^{-1} = 3^{-1} \cdot 2^{-1}$

So we have: $\frac{3^5 \cdot 2^3}{3^{-1} \cdot 2^{-1}} = 3^{5-(-1)} \cdot 2^{3-(-1)} = 3^6 \cdot 2^4$ Now calculate the final result: $3^6 = 729 \quad \text{and} \quad 2^4 = 16$ Thus, the final result is: $729 \cdot 16 = 11664$

Summary:

1. $a^6b^{-10}c^{12}$
2. $x^{-5}y^5z^{-4}c^3$
3. $11664$

Would you like any further details on the steps?

Here are 5 related questions to explore:

1. How do we handle negative exponents in fractions?
2. What happens when applying a negative exponent to a simplified term?
3. How does the distributive property apply to exponents across multiplication?
4. Can you explain how to simplify more complex algebraic expressions with multiple variables?
5. What are the common mistakes when simplifying expressions with negative exponents?

Tip: When simplifying expressions with multiple terms, handle one variable at a time to avoid confusion.