Multiply :
`3 - (2)/(3)xy + (5)/(7)xy^(2) - (16)/(21)x^(2)y` by `-21 x^(2)y^(2)`

Answer

To solve the problem of multiplying the expression \(3 - \frac{2}{3}xy + \frac{5}{7}xy^2 - \frac{16}{21}x^2y\) by \(-21x^2y^2\), we will follow these steps:

### Step 1: Write the expression to be multiplied
We start with the expression:
\[ 3 - \frac{2}{3}xy + \frac{5}{7}xy^2 - \frac{16}{21}x^2y \]
and we will multiply this by \(-21x^2y^2\).

### Step 2: Distribute \(-21x^2y^2\) to each term
We will multiply each term in the expression by \(-21x^2y^2\):

1. **First term**:
\[ 3 \cdot (-21x^2y^2) = -63x^2y^2 \]

2. **Second term**:
\[ -\frac{2}{3}xy \cdot (-21x^2y^2) = \frac{2 \cdot 21}{3} x^{1+2} y^{1+2} = \frac{42}{3} x^3y^3 = 14x^3y^3 \]

3. **Third term**:
\[ \frac{5}{7}xy^2 \cdot (-21x^2y^2) = -\frac{5 \cdot 21}{7} x^{1+2} y^{2+2} = -15x^3y^4 \]

4. **Fourth term**:
\[ -\frac{16}{21}x^2y \cdot (-21x^2y^2) = \frac{16 \cdot 21}{21} x^{2+2} y^{1+2} = 16x^4y^3 \]

### Step 3: Combine all the terms
Now we combine all the results from the multiplications:
\[ -63x^2y^2 + 14x^3y^3 - 15x^3y^4 + 16x^4y^3 \]

### Step 4: Write the final answer
The final expression after combining all the terms is:
\[ -63x^2y^2 + 14x^3y^3 - 15x^3y^4 + 16x^4y^3 \]

### Summary of the final expression:
\[ -63x^2y^2 + 14x^3y^3 - 15x^3y^4 + 16x^4y^3 \]

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