Multiply: `(-5x^(2)y),((-2)/(3)xy^(2)z),((8)/(15)xyz^(3))` and `((-1)/(4)z)`

Answer

To solve the problem of multiplying the expressions \((-5x^2y)\), \(\left(-\frac{2}{3}xy^2z\right)\), \(\left(\frac{8}{15}xyz^3\right)\), and \(\left(-\frac{1}{4}z\right)\), we will follow these steps:

### Step-by-Step Solution:

1. **Multiply the First Two Expressions:**
\[ (-5x^2y) \cdot \left(-\frac{2}{3}xy^2z\right) \]
- Multiply the constants:
\[ -5 \cdot -\frac{2}{3} = \frac{10}{3} \quad (\text{since } - \times - = +) \]
- Multiply the variables:
- For \(x^2 \cdot x\):
\[ x^{2+1} = x^3 \]
- For \(y \cdot y^2\):
\[ y^{1+2} = y^3 \]
- \(z\) remains as is.
- Combine the results:
\[ \frac{10}{3}x^3y^3z \]

2. **Multiply the Result with the Third Expression:**
\[ \left(\frac{10}{3}x^3y^3z\right) \cdot \left(\frac{8}{15}xyz^3\right) \]
- Multiply the constants:
\[ \frac{10}{3} \cdot \frac{8}{15} = \frac{80}{45} = \frac{16}{9} \quad (\text{after simplifying}) \]
- Multiply the variables:
- For \(x^3 \cdot x\):
\[ x^{3+1} = x^4 \]
- For \(y^3 \cdot y\):
\[ y^{3+1} = y^4 \]
- For \(z \cdot z^3\):
\[ z^{1+3} = z^4 \]
- Combine the results:
\[ \frac{16}{9}x^4y^4z^4 \]

3. **Multiply the Result with the Fourth Expression:**
\[ \left(\frac{16}{9}x^4y^4z^4\right) \cdot \left(-\frac{1}{4}z\right) \]
- Multiply the constants:
\[ \frac{16}{9} \cdot -\frac{1}{4} = -\frac{16}{36} = -\frac{4}{9} \quad (\text{after simplifying}) \]
- Multiply the variables:
- \(x^4\) remains as is.
- \(y^4\) remains as is.
- For \(z^4 \cdot z\):
\[ z^{4+1} = z^5 \]
- Combine the results:
\[ -\frac{4}{9}x^4y^4z^5 \]

### Final Answer:
\[ -\frac{4}{9}x^4y^4z^5 \]