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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 expressions 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:
Step 1: Find the product of the numerators \(6a\) and \(b^3\).
\(6a\) \(\times\) \(b^3\) \(=\) \(6ab^3\)
Step 2: Find the product of the denominators \(b^2\) and \(3a^2\).
\(b^2\) \(\times\) \(3a^2\) \(=\) \(3a^2b^2\)
Step 3: Reduce the expression \(\frac{6ab^3}{3a^2b^2}\) to its lowest form.
\(\frac{6ab^3}{3a^2b^2}\) \(=\) \(\frac{\not{3} \times 2 \times \not{a} \times \not{b^2} \times b}{\not{3} \times \not{a} \times a \times \not{b^2}}\)
\(=\) \(\frac{2b}{a}\)
Step 4: Write the resulting expression.
\(\frac{6a}{b^2}\) \(\times\) \(\frac{b^3}{3a^2}\) \(=\) \(\frac{2b}{a}\)
Therefore, the product of \(\frac{6a}{b^2}\) and \(\frac{b^3}{3a^2}\) is \(\frac{2b}{a}\).
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