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The rational expressions do support the division operation.

Let us learn how to divide two rational expressions.

*Division of Rational Expressions*

The division of one rational expression by the other is equivalent to the product of the first and reciprocal of the second expression.

**:**

*Working rule to divide 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 ratio of these two rational numbers is computed as follows:

**Step 1**: Find the reciprocal of the second expression \(\frac{s(x)}{t(x)}\).

**Step 2**: Find the product of \(\frac{p(x)}{q(x)}\) and \(\frac{t(x)}{s(x)}\).

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

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

Example:

Divide \(\frac{6a}{b^2}\) by \(\frac{a^3}{3b^2}\).

**Solution**:

**: Find the reciprocal of \(\frac{a^3}{3b^2}\).**

*Step 1*\(\frac{1}{\frac{a^3}{3b^2}}\) \(=\) \(\frac{3b^2}{a^3}\).

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

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

\(=\) \(\frac{18ab^2}{a^3b^2}\)

**: Reduce the expression \(\frac{18ab^2}{a^3b^2}\) to its lowest form.**

*Step 3*\(\frac{18ab^2}{a^3b^2}\) \(=\) \(\frac{18 \times \not{a} \times \not{b^2}}{\not{a} \times a^2 \times \not{b^2}}\)

\(=\) \(\frac{18}{a^2}\)

**: Write the resulting expression.**

*Step 4*\(\frac{6a}{b^2}\) \(\div\) \(\frac{a^3}{3b^2}\) \(=\) \(\frac{18}{a^2}\)

Therefore, \(\frac{6a}{b^2}\) \(\div\) \(\frac{a^3}{3b^2}\) is \(\frac{18}{a^2}\).