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### Theory:

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 other is equivalent to the product of 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:

Step 1: Find the reciprocal of $$\frac{a^3}{3b^2}$$.

$$\frac{1}{\frac{a^3}{3b^2}}$$ $$=$$ $$\frac{3b^2}{a^3}$$.

Step 2: Find the product of $$\frac{6a}{b^2}$$ and $$\frac{3b^2}{a^3}$$.

$$\frac{6a}{b^2}$$ $$\times$$ $$\frac{3b^2}{a^3}$$ $$=$$ $$\frac{6a \times 3b^2}{b^2 \times a^3}$$

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

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

$$\frac{18ab^2}{a^3b^2}$$ $$=$$ $$\frac{18 \times \not{a} \times \not{b^2}}{\not{a} \times a^2 \times \not{b^2}}$$

$$=$$ $$\frac{18}{a^2}$$

Step 4: Write the resulting expression.

$$\frac{6a}{b^2}$$ $$\div$$ $$\frac{a^3}{3b^2}$$ $$=$$ $$\frac{18}{a^2}$$

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