To multiply the fractions \(-\frac{6}{5} \times -\frac{3}{25} \times -\frac{6}{50} \times \frac{3}{5}\), we will follow these steps:
- Multiply the numerators together.
- Multiply the denominators together.
- Simplify the resulting fraction.
- Consider the signs.
Step 1: Multiply the Numerators
Numerators:
- \(-6\)
- \(-3\)
- \(-6\)
- \(3\)
\[ -6 \times -3 \times -6 \times 3 \]
Calculate: \[ (-6) \times (-3) = 18 \quad (\text{negative times negative is positive}) \] \[ 18 \times (-6) = -108 \quad (\text{positive times negative is negative}) \] \[ -108 \times 3 = -324 \quad (\text{negative times positive is negative}) \]
So, the product of the numerators is \(-324\).
Step 2: Multiply the Denominators
Denominators:
- \(5\)
- \(25\)
- \(50\)
- \(5\)
\[ 5 \times 25 \times 50 \times 5 \]
Calculate: \[ 5 \times 25 = 125 \] \[ 125 \times 50 = 6250 \] \[ 6250 \times 5 = 31250 \]
So, the product of the denominators is \(31250\).
Step 3: Combine the Results
Now we combine our results from the numerators and denominators:
\[ -\frac{324}{31250} \]
Step 4: Simplify
To simplify \(-\frac{324}{31250}\), we find the greatest common divisor (GCD) of \(324\) and \(31250\).
- \(324 = 2^2 \times 3^4\)
- \(31250 = 2 \times (5^5)\)
Since their only common factor is \(2\), we can divide both the numerator and the denominator by \(2\):
\[ -\frac{324 \div 2}{31250 \div 2} = -\frac{162}{15625} \]
Final Answer
The final answer in simplest form is:
\[ -\frac{162}{15625} \]