Multiply `-5xy^(2), -3x^(2)yz` and `8yz^(2)`. Verify the result for x = 1, y = 2 and z = 3.

Answer

To solve the problem of multiplying the algebraic expressions \(-5xy^2\), \(-3x^2yz\), and \(8yz^2\), we will follow these steps:

### Step 1: Write down the expressions
We have three expressions to multiply:
1. \(-5xy^2\)
2. \(-3x^2yz\)
3. \(8yz^2\)

### Step 2: Multiply the coefficients
First, we will multiply the coefficients (the numerical parts) of the expressions:
- The coefficients are \(-5\), \(-3\), and \(8\).
- Multiply them:
\[ (-5) \times (-3) \times 8 = 15 \times 8 = 120 \]

### Step 3: Multiply the variables
Next, we will multiply the variable parts:
- For \(x\):
\[ x^1 \text{ (from } -5xy^2\text{)} \text{ and } x^2 \text{ (from } -3x^2yz\text{)} \implies x^{1+2} = x^3 \]
- For \(y\):
\[ y^2 \text{ (from } -5xy^2\text{)} \text{ and } y^1 \text{ (from } -3x^2yz\text{)} \implies y^{2+1} = y^3 \]
- For \(z\):
\[ z^1 \text{ (from } -3x^2yz\text{)} \text{ and } z^2 \text{ (from } 8yz^2\text{)} \implies z^{1+2} = z^3 \]

### Step 4: Combine the results
Now, we combine the results from steps 2 and 3:
\[ 120x^3y^3z^3 \]

### Step 5: Verify the result for \(x = 1\), \(y = 2\), and \(z = 3\)
Now we will substitute \(x = 1\), \(y = 2\), and \(z = 3\) into the expression:
\[ 120(1^3)(2^3)(3^3) \]
Calculating each part:
- \(1^3 = 1\)
- \(2^3 = 8\)
- \(3^3 = 27\)

Now substituting these values:
\[ 120 \times 1 \times 8 \times 27 \]
Calculating:
\[ 120 \times 8 = 960 \]
Then,
\[ 960 \times 27 = 25920 \]

### Final Result
The final result of the multiplication and verification is:
\[ \text{Result} = 25920 \]