# Multiplying algebraic fractions

In this chapter we will learn to multiply algebraic fractions with the help of solved examples.

To understand the chapter, you should have basic knowledge of multiplication of algebraic expressions.

## How to multiply algebraic fractions ?

Multiplication of algebraic fractions is similar to multiplication of general fraction number.

The only difference here is that you have to multiply the algebraic expressions.

Two or more algebraic expressions can be multiplied by multiplying the numerator and denominator separately and then simplifying the fraction ( if possible).

Generally, the multiplication can be expressed as;

\mathtt{\Longrightarrow \frac{Numerator\ 1}{Denominator\ 1} \times \ \frac{Numerator\ 2}{Denominator\ 2}}\\\ \\ \mathtt{\Longrightarrow \frac{Numerator\ 1\ \times Numerator\ 2}{Denominator\ 1\ \times Denominator\ 2}}

I hope you understood the above concept. Let us see some solved examples for further understanding.

## Multiplying algebraic fractions – Solved examples

Example 01
Multiply the fractions.

\mathtt{\Longrightarrow \frac{3x}{5} \times \ \frac{2xy}{7}}

Solution
\mathtt{\Longrightarrow \ \frac{3x\times 2xy}{5\times 7}}\\\ \\ \mathtt{\Longrightarrow \frac{6x^{2} y}{35}}

Example 02
Multiply the fractions
\mathtt{\Longrightarrow \frac{x}{3} \times \ \frac{x+4}{5}}

Solution

\mathtt{\Longrightarrow \frac{x}{3} \times \ \frac{x+4}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x( x+4)}{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \frac{x^{2} +4x}{15}}

Example 03
Multiply the algebraic fractions.

\mathtt{\Longrightarrow \frac{9x^{2}}{4y} \times \ \frac{3( x+y)}{3x} \times \frac{2x^{3} y}{x^{2} y^{3}}}

Solution

\mathtt{\Longrightarrow \ \frac{9x^{2} \ \times \ 3( x+y) \ \times \ 2x^{3} y}{4y\times \ 3x\times \ x^{2} y^{3}}}\\\ \\ \mathtt{\Longrightarrow \frac{9\times 3\times 2\times \ x^{2+3} .y.( x+y)}{4\times 3\times x^{2+1} .\ y^{3+1}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{54.\ x^{5} .y.( x+y) \ }{12.x^{3} .y^{4}}}\

Example 04
Multiply the algebraic expression.

\mathtt{\Longrightarrow \frac{x+6}{x^{3}} \ \times \ \frac{4x}{x^{2} -36}}

Solution

\mathtt{\Longrightarrow \frac{x+6}{x^{3}} \ \times \ \frac{4x}{x^{2} -36}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x\ +6}{x^{3}} \times \frac{4x}{( x-6)( x+6)}}

Cancelling out common terms from numerator and denominator.

\mathtt{\Longrightarrow \ \frac{\cancel{( x+6)}}{x^{3-1}} \times \frac{4\cancel{x}}{( x-6)\cancel{( x+6)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{x^{2}( x-6)}}

Example 05
Multiply the algebraic fractions.

\mathtt{\Longrightarrow \frac{x^{2} -25}{x-7} \ \times \ \frac{3x-21}{x^{2} +10x+25}}

Solution

\mathtt{\Longrightarrow \ \frac{x^{2} -5^{2}}{x-7} \times \frac{2( x-7)}{x^{2} +2.x.5+5^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( x-5)( x+5)}{( x-7)} \ \times \ \ \frac{2( x-7)}{( x+5)^{2}}}

Cancelling out common terms in numerator and denominator.

\mathtt{\Longrightarrow \ \frac{( x-5)\cancel{( x+5)}}{\cancel{( x-7)}} \ \times \ \ \frac{2\ \cancel{( x-7)}}{( x+5)^{\cancel{2}}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2( x-5)}{( x+5)}}

Example 06
Multiply the algebraic fractions.

\mathtt{\Longrightarrow \frac{5}{x-7} \ \times \ \frac{10xy}{( x+7)} \times \frac{( x+y)}{3x}}

Solution

\mathtt{\Longrightarrow \ \frac{5\times \ 10\times xy.( x+y) \ }{3x.( x-7)( x+7)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{50.\cancel{x} .y.( x+y)}{3.\cancel{x} .\left( x^{2} -7^{2}\right)} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{50y( x+y)}{3\ \left( x^{2} -7^{2}\right)}}

Example 07
Multiply the algebraic fractions.

\mathtt{\Longrightarrow \frac{x\ ( x+2)}{7y} \times \frac{15x^{2} y^{3}}{\left( x^{2} -4\right)}}

Solution

\mathtt{\Longrightarrow \ \frac{x\ ( x+2)}{7\cancel{y}} \times \frac{15x^{2} y^{3-1}}{\left( x^{2} -2^{2}\right)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x( x+2)}{7} \times \frac{15x^{2} y^{2}}{( x-2)( x+2)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x\cancel{( x+2)}}{7} \times \frac{15x^{2} y^{2}}{( x-2)\cancel{( x+2)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{15x^{3} y^{2}}{7( x-2)}}