Certainly! Let’s work through this problem step by step.
1. Rewrite Mixed Numbers as Improper Fractions:
- The expression we are given is: [tex]\(-\frac{2}{3} \cdot 2 \frac{2}{5} \cdot 9 \cdot -3 \frac{1}{3}\)[/tex].
- First, convert the mixed numbers to improper fractions:
- [tex]\(2 \frac{2}{5} = \frac{2 \cdot 5 + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5}\)[/tex]
- [tex]\(-3 \frac{1}{3} = -\frac{3 \cdot 3 + 1}{3} = -\frac{9 + 1}{3} = -\frac{10}{3}\)[/tex]
2. Rewrite the Expression:
- Replacing the mixed numbers with their corresponding improper fractions, we get:
[tex]\[
-\frac{2}{3} \cdot \frac{12}{5} \cdot 9 \cdot -\frac{10}{3}
\][/tex]
3. Use the Commutative Property of Multiplication:
- The Commutative Property of Multiplication states that the product of a set of numbers is the same regardless of the order in which they are multiplied. Therefore, we can rearrange the numbers:
[tex]\[
\left(-\frac{2}{3}\right) \cdot \left(-\frac{10}{3}\right) \cdot \frac{12}{5} \cdot 9
\][/tex]
- However, for simplicity, I'll keep the order as is and proceed to evaluate.
4. Evaluate Step-by-Step:
- First, multiply the fractions. Let's break it down:
- Calculate [tex]\(-\frac{2}{3} \cdot \frac{12}{5}\)[/tex]:
[tex]\[
-\frac{2 \cdot 12}{3 \cdot 5} = -\frac{24}{15} = -\frac{8}{5} \quad \text{(simplified the fraction)}
\][/tex]
- Then multiply the result by 9:
[tex]\[
-\frac{8}{5} \cdot 9 = -\frac{8 \cdot 9}{5} = -\frac{72}{5}
\][/tex]
- Now, multiply by [tex]\(-\frac{10}{3}\)[/tex]:
[tex]\[
-\frac{72}{5} \cdot -\frac{10}{3} = \frac{72 \cdot 10}{5 \cdot 3} = \frac{720}{15} = 48 \quad \text{(simplified the fraction)}
\][/tex]
5. Final Result:
- After evaluating the multiplication step-by-step, we find the result is [tex]\(48.0\)[/tex].
Therefore, the evaluated expression is [tex]\(48.0\)[/tex].
