To multiply the given rational expressions, follow these step-by-step instructions:
Given:
[tex]\[
\frac{14 x^2 y}{5 y^2 x} \cdot \frac{25 x^3 y^2}{7 x^2 y^3}
\][/tex]
### Step 1: Multiply the Numerators
First, we multiply the numerators of the two fractions:
[tex]\[
14 x^2 y \cdot 25 x^3 y^2
\][/tex]
Multiply each term:
[tex]\[
14 \cdot 25 = 350
\][/tex]
[tex]\[
x^2 \cdot x^3 = x^{2+3} = x^5
\][/tex]
[tex]\[
y \cdot y^2 = y^{1+2} = y^3
\][/tex]
So the numerator of the product is:
[tex]\[
350 x^5 y^3
\][/tex]
### Step 2: Multiply the Denominators
Next, we multiply the denominators of the two fractions:
[tex]\[
5 y^2 x \cdot 7 x^2 y^3
\][/tex]
Multiply each term:
[tex]\[
5 \cdot 7 = 35
\][/tex]
[tex]\[
y^2 \cdot y^3 = y^{2+3} = y^5
\][/tex]
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So the denominator of the product is:
[tex]\[
35 x^3 y^5
\][/tex]
### Step 3: Write the Product of the Rational Expressions
Combine the results from Steps 1 and 2 into a single fraction:
[tex]\[
\frac{350 x^5 y^3}{35 x^3 y^5}
\][/tex]
### Step 4: Simplify the Fraction
Now, simplify the fraction by cancelling out common factors in the numerator and the denominator.
Divide the coefficients (numerical parts):
[tex]\[
\frac{350}{35} = 10
\][/tex]
Reduce the powers of [tex]\( x \)[/tex]:
[tex]\[
\frac{x^5}{x^3} = x^{5-3} = x^2
\][/tex]
Reduce the powers of [tex]\( y \)[/tex]:
[tex]\[
\frac{y^3}{y^5} = y^{3-5} = y^{-2} = \frac{1}{y^2}
\][/tex]
### Step 5: Combine the Results
So, the simplified expression is:
[tex]\[
10 \cdot x^2 \cdot \frac{1}{y^2} = \frac{10 x^2}{y^2}
\][/tex]
### Final Answer
The product of the given rational expressions, simplified, is:
[tex]\[
\boxed{\frac{10 x^2}{y^2}}
\][/tex]
