Find the product `(3/5abc^2)*(-25/12a^2b^2)*(-8b^2c)`.

Answer

To find the product \((\frac{3}{5}abc^2) \cdot (-\frac{25}{12}a^2b^2) \cdot (-8b^2c)\), we will follow these steps:

### Step 1: Write the expression clearly
We start with the expression:
\[ \left(\frac{3}{5}abc^2\right) \cdot \left(-\frac{25}{12}a^2b^2\right) \cdot \left(-8b^2c\right) \]

### Step 2: Combine the coefficients
First, we will multiply the numerical coefficients:
\[ \frac{3}{5} \cdot -\frac{25}{12} \cdot -8 \]

### Step 3: Simplify the coefficients
Calculating step by step:
1. Multiply \(-\frac{25}{12}\) and \(-8\):
\[ -\frac{25}{12} \cdot -8 = \frac{25 \cdot 8}{12} = \frac{200}{12} \]
Now simplify \(\frac{200}{12}\):
\[ \frac{200 \div 4}{12 \div 4} = \frac{50}{3} \]

2. Now multiply \(\frac{3}{5}\) with \(\frac{50}{3}\):
\[ \frac{3}{5} \cdot \frac{50}{3} = \frac{3 \cdot 50}{5 \cdot 3} = \frac{50}{5} = 10 \]

### Step 4: Combine the variables
Now we will combine the variable parts:
- From \(\frac{3}{5}abc^2\), we have \(a^1, b^1, c^2\).
- From \(-\frac{25}{12}a^2b^2\), we have \(a^2, b^2\).
- From \(-8b^2c\), we have \(b^2, c^1\).

Now we add the powers of like bases:
- For \(a\):
\[ a^{1+2} = a^3 \]
- For \(b\):
\[ b^{1+2+2} = b^5 \]
- For \(c\):
\[ c^{2+1} = c^3 \]

### Step 5: Combine everything
Putting it all together, we have:
\[ 10a^3b^5c^3 \]

### Final Answer
Thus, the product is:
\[ \boxed{10a^3b^5c^3} \]