### Theory:

The rational expressions do support the multiplication operation.

Let us learn how to multiply two rational expressions.

*Multiplication of Rational Expressions*

The product of two rational expression is the product of their numerators divided by the product of their denominators.

**:**

*Working rule to find the product of two rational expressions*Let \(\frac{p(x)}{q(x)}\) and \(\frac{s(x)}{t(x)}\) are two rational numbers where \(q(x) \neq 0\) and \(t(x) \neq 0\), then the product of these two rational numbers is computed as follows:

**Step 1**: Find the product of the numerators \(p(x)\) and \(s(x)\).

That is \(p(x)\) \(\times\) \(s(x)\).

**Step 2**: Find the product of the denominators \(q(x)\) and \(t(x)\).

That is \(q(x)\) \(\times\) \(t(x)\).

**Step 3**: Reduce the expression \(\frac{p(x) \times s(x)}{q(x) \times t(x)}\) to its lowest form.

**Step 4**: The resulting expression obtained is the product of the given rational expression.

Example:

Find the product of \(\frac{6a}{b^2}\) and \(\frac{b^3}{3a^2}\).

**Solution**:

**: Find the product of the numerators \(6a\) and \(b^3\).**

*Step 1*\(6a\) \(\times\) \(b^3\) \(=\) \(6ab^3\)

**: Find the product of the denominators \(b^2\) and \(3a^2\).**

*Step 2*\(b^2\) \(\times\) \(3a^2\) \(=\) \(3a^2b^2\)

**: Reduce the expression \(\frac{6ab^3}{3a^2b^2}\) to its lowest form.**

*Step 3*\(\frac{6ab^3}{3a^2b^2}\) \(=\) \(\frac{\not{3} \times 2 \times \not{a} \times \not{b^2} \times b}{\not{3} \times \not{a} \times a \times \not{b^2}}\)

\(=\) \(\frac{2b}{a}\)

**: Write the resulting expression.**

*Step 4*\(\frac{6a}{b^2}\) \(\times\) \(\frac{b^3}{3a^2}\) \(=\) \(\frac{2b}{a}\)

Therefore, the product of \(\frac{6a}{b^2}\) and \(\frac{b^3}{3a^2}\) is \(\frac{2b}{a}\).