To solve the given problem and reduce it if possible, we need to follow a series of steps involving conversion, multiplication, and simplification. Let's break it down step by step:
1. Convert the mixed numbers to improper fractions:
- The first mixed number is [tex]\(-4 \frac{1}{11}\)[/tex]. To convert this to an improper fraction, we proceed as follows:
[tex]\[
-4 \frac{1}{11} = -4 - \frac{1}{11} = -\left(4 + \frac{1}{11}\right)
= -\left(\frac{4 \times 11 + 1}{11}\right)
= -\left(\frac{44 + 1}{11}\right)
= -\left(\frac{45}{11}\right)
= -\frac{45}{11}
\][/tex]
- The second mixed number is [tex]\(-2 \frac{5}{6}\)[/tex]. To convert this to an improper fraction, we proceed as follows:
[tex]\[
-2 \frac{5}{6} = -2 - \frac{5}{6} = -\left(2 + \frac{5}{6}\right)
= -\left(\frac{2 \times 6 + 5}{6}\right)
= -\left(\frac{12 + 5}{6}\right)
= -\left(\frac{17}{6}\right)
= -\frac{17}{6}
\][/tex]
2. Multiply the improper fractions:
- Now, we need to multiply [tex]\(-\frac{45}{11}\)[/tex] and [tex]\(-\frac{17}{6}\)[/tex]:
[tex]\[
\left(-\frac{45}{11}\right) \times \left(-\frac{17}{6}\right)
\][/tex]
- Multiplying the numerators and denominators together, we get:
[tex]\[
\left(-\frac{45 \cdot 17}{11 \cdot 6}\right) = \left(\frac{765}{66}\right)
\][/tex]
3. Simplify the fraction if possible:
- The product we obtained is [tex]\(\frac{765}{66}\)[/tex]. To simplify this fraction, we find the greatest common divisor (GCD) of the numerator and the denominator:
[tex]\[
\text{GCD}(765, 66) = 3
\][/tex]
- Dividing both the numerator and the denominator by their GCD, we get:
[tex]\[
\frac{765 \div 3}{66 \div 3} = \frac{255}{22}
\][/tex]
- Thus, [tex]\(\frac{255}{22}\)[/tex] is the simplified form of the product.
Final Answer:
Therefore, the product of [tex]\(\left(-4 \frac{1}{11}\right) \cdot\left(-2 \frac{5}{6}\right)\)[/tex] in its simplest form is:
[tex]\[
\frac{255}{22}
\][/tex]
