Sure! Let's solve this step-by-step.
Given the expression to multiply:
[tex]\[ 4 \cdot \frac{6}{-7} \cdot \frac{-5}{3} \][/tex]
First, simplify the multiplication of fractions:
[tex]\[ \frac{6}{-7} = -\frac{6}{7} \][/tex]
[tex]\[ \frac{-5}{3} = -\frac{5}{3} \][/tex]
Thus, the expression can be rewritten as:
[tex]\[ 4 \cdot -\frac{6}{7} \cdot -\frac{5}{3} \][/tex]
Next, multiply the integers and the fractions separately:
[tex]\[ 4 \cdot -\frac{6}{7} = -4 \cdot \frac{6}{7} = -\frac{24}{7} \][/tex]
Now, multiply [tex]\(-\frac{24}{7}\)[/tex] with [tex]\(-\frac{5}{3}\)[/tex]:
[tex]\[ -\frac{24}{7} \cdot -\frac{5}{3} \][/tex]
The multiplication of two negative fractions yields a positive result:
[tex]\[ -\frac{24}{7} \cdot -\frac{5}{3} = \frac{24 \cdot 5}{7 \cdot 3} = \frac{120}{21} \][/tex]
Finally, we need to simplify [tex]\(\frac{120}{21}\)[/tex].
The greatest common divisor (GCD) of [tex]\(120\)[/tex] and [tex]\(21\)[/tex] is [tex]\(3\)[/tex].
So, divide both the numerator and the denominator by [tex]\(3\)[/tex]:
[tex]\[ \frac{120 \div 3}{21 \div 3} = \frac{40}{7} \][/tex]
Thus, the simplest form of the product is [tex]\(\frac{40}{7}\)[/tex].
In decimal form:
[tex]\[ \frac{40}{7} \approx 5.714285714285714 \][/tex]
So, the final answers are:
- In simplest fraction form: [tex]\(\frac{40}{7}\)[/tex]
- In decimal form: [tex]\(5.714285714285714\)[/tex]
- As a precise fraction: [tex]\(\frac{6433713753386423}{1125899906842624}\)[/tex]
This [tex]\(\frac{6433713753386423}{1125899906842624}\)[/tex] is a more precise representation of the simplified product.
