Multiply :
`3/7 x^(2) y^(3) and -14/15xy^(2)`

Answer

To solve the problem of multiplying the algebraic expressions \( \frac{3}{7} x^{2} y^{3} \) and \( -\frac{14}{15} xy^{2} \), we can follow these steps:

### Step 1: Multiply the numerical coefficients
We start by multiplying the numerical parts of the expressions:
\[ \frac{3}{7} \times -\frac{14}{15} \]
To do this, we multiply the numerators and the denominators:
\[ = \frac{3 \times -14}{7 \times 15} \]
Calculating the numerator:
\[ 3 \times -14 = -42 \]
Calculating the denominator:
\[ 7 \times 15 = 105 \]
So we have:
\[ \frac{-42}{105} \]

### Step 2: Simplify the numerical coefficient
Now we simplify \( \frac{-42}{105} \). We can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 21:
\[ \frac{-42 \div 21}{105 \div 21} = \frac{-2}{5} \]

### Step 3: Multiply the variables
Next, we multiply the variable parts. We have:
\[ x^{2} \times x^{1} \quad \text{and} \quad y^{3} \times y^{2} \]
Using the property of exponents that states \( x^{a} \times x^{b} = x^{a+b} \), we can add the exponents:
\[ x^{2+1} = x^{3} \quad \text{and} \quad y^{3+2} = y^{5} \]

### Step 4: Combine the results
Now we combine the simplified numerical coefficient with the variable parts:
\[ \frac{-2}{5} x^{3} y^{5} \]

### Final Answer
Thus, the final result of multiplying \( \frac{3}{7} x^{2} y^{3} \) and \( -\frac{14}{15} xy^{2} \) is:
\[ \frac{-2}{5} x^{3} y^{5} \]

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