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Multiplying and dividing algebraic fractions

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To the algebraic fractions are fractions in which they appear polynomials in the numerator and denominator or at least in the denominator.

Examples:

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\dpi{120} \mathrm{\frac{2x}{5y}}\dpi{120} \mathrm{\frac{x-1}{2y^2}}\dpi{120} \mathrm{\frac{a-b}{a^2-b^2}}\dpi{120} \mathrm{\frac{1}{x^3 -8}}

Thus, the multiplication and division of algebraic fractions involves calculations between polynomials, that is, it involves operations between terms with one or more variables.

Multiplying Algebraic Fractions

A multiplying algebraic fractions is similar to multiplying numerical fractions.

Just multiply the numerators together and multiply the denominators together.

Remember that in multiplication of powers If the bases are the same, keep the base and add the exponents: \dpi{120} \mathrm{x^n.x^m x^{n+ m}}.

Examples:

a) Calculate \dpi{120} \mathrm{\frac{x^3}{3y}\cdot \frac{5x^2}{2y^3}}.

\dpi{120} \mathrm{\frac{x^3}{3y}\cdot \frac{5x^2}{2y^3} \frac{x^3\cdot 5x^2}{3y\cdot 2y^ 3} \frac{5x^{5}}{6y^4}}

b) Calculate \dpi{120} \mathrm{\frac{xy}{a^2b}\cdot \frac{a}{2x}}.

\dpi{120} \mathrm{\frac{xy}{a^2b}\cdot \frac{a}{2x} \frac{\cancel{\mathrm{x}}\cdot y\cdot \cancel{\mathrm {a}}}{a^{\cancel{2}}\cdot b\cdot 2\cdot \cancel{\mathrm{x}}} \frac{y}{2ab}}

Note that when we do multiplication, we can simplify the algebraic fraction by canceling the equal factors.

Division of algebraic fractions

A division of algebraic fractions is similar to division of numerical fractions

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. Just keep the first fraction and multiply by the reciprocal of the second fraction.

The reciprocal of the second fraction is obtained by switching the numerator and denominator around.

Examples:

a) Calculate \dpi{120} \mathrm{\frac{3x}{8y}:\frac{x^5}{4y}}.

Keeping the first fraction and multiplying by the reciprocal of the second, we have:

\dpi{120} \mathrm{\frac{3x}{8y}:\frac{x^5}{4y} \frac{3x}{8y}\cdot \frac{4y}{x^5} }

So, we just need to solve this multiplication between fractions:

\dpi{120} \mathrm{ \frac{3x}{8y}\cdot \frac{4y}{x^5} \frac{12xy}{8x^5y} \frac{3}{2x^4} }

Therefore, the result of the division is:

\dpi{120} \mathrm{\frac{3x}{8y}:\frac{x^5}{4y} \frac{3}{2x^4}}

b) Calculate \dpi{120} \mathrm{\frac{a}{b+1}:\frac{a^4}{b^2-1}}.

Keeping the first fraction and multiplying by the reciprocal of the second, we have:

\dpi{120} \mathrm{\frac{a}{b+1}:\frac{a^4}{b^2-1} \frac{a}{b+1}\cdot \frac{b^ 2-1}{a^4} }

Now, we solve the multiplication between fractions:

\dpi{120} \mathrm{ \frac{a}{b+1}\cdot \frac{b^2-1}{a^4} \frac{a\cdot (b^2-1)}{a ^4\cdot (b+1)} \frac{\cancel{\mathrm{a}}\cdot (b-1)\cdot \cancel{(\mathrm{b+1})}}{a^{\cancel{4}}\cdot \cancel{ (\mathrm{b+1})}} \frac{b-1}{a^3}}

For simplicity, in the second equality, we use the factoring the difference of two squares.

Therefore, the result of the division is:

\dpi{120} \mathrm{\frac{a}{b+1}:\frac{a^4}{b^2-1} \frac{b-1}{a^3}}

You may also be interested:

  • List of fraction multiplication exercises
  • List of fraction division exercises
  • List of Factoring Exercises
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