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Simplification of Rational Expressions

Subject: Compulsory Maths

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This note includes the information about the multiplication and division of Rational Expressions.
Simplification of Rational Expressions

Simplification of Rational Expression

source; slideplayer.com Fig: Simplification of Rational Expression
source; slideplayer.com
Fig: Simplification of Rational Expression

As we know that 1, 2, \(\frac{1}{2}\),\(\frac{-4}{5}\), etc are the rational numbers. Similarly,\(\frac{x}{2}\),\(\frac{x^2}{y^2}\),\(\frac{x+3}{x-8}\), etc. are called rational expressions. Here we shall discuss about addition, subtraction,multiplication and division of rational expressions.

Multiplication of Rational Expressions

source: ctle.hccs.edu Fig: Multiplication of Rational
source: ctle.hccs.edu
Fig: Multiplication of Rational Expression

In multiplication, we simplify the numerical coefficients as in a case of multiplication of fraction. In the case of variables, we apply the product and quotient rules of indices. In a division, we should multiply the dividend by the reciprocal of a division. For example,

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\(\frac{4a^3b^2}{5a^3y^3}\)×\(\frac{10x^4y^3}{12a^4b}\)

\(\frac{2x^{4-3}× y^{3-2}}{3a^{4-3}×b^{3-2}}\)

\(\frac{2xy}{3ab}\)ans.

Addition and Subtraction of Rational Expression with the different denominators

In this case, we should find the LCM of the denominators. Then the LCM is divided by each denominator and the quotient is multiplied by the corresponding numerator as in a case of simplification of unlike fraction. For example,

\(\frac{x}{2}\) +\(\frac{x}{6}\)

\(\frac{3x+x}{6}\) =\(\frac{4x}{6}\) =\(\frac{2x}{3}\) ans.

Things to remember
  •  In a division, we should multiply the dividend by the reciprocal of a division.
  • LCM is divided by each denominator and the quotient is multiplied by the corresponding numerator as in a case of simplification of unlike fraction.
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Videos for Simplification of Rational Expressions
Simplify Complex Rational Expressions
Simplifying Rational Expressions
Questions and Answers

Solution:

\(\frac{4a^3b^2}{5x^3y^2}\) × \(\frac{10x^4y^3}{12a^4b^3}\) 

 = \(\frac{4a^3b^2×10x^4y^3}{5x^3y^2×12a^4b^3}\)

 = \(\frac{2x^{4-3}×y^{3-2}}{3a^{4-3}b^{3-2}}\)

 = \(\frac{2xy}{3ab}\)

Solution:

 = \(\frac{5x^2}{8ax^3}\) ÷ \(\frac{15a^3y^3}{16a^2x^2}\)

 = \(\frac{5x^2}{8ax^3}\) × \(\frac{15a^3y^3}{16a^2x^2}\)

 = \(\frac{2a^2x^4y}{3a^4x^3y^3}\)

 = \(\frac{2x^{4-3}}{3a^{4-2}y^{3-1}}\)

 = \(\frac{2x}{3a^2y^2}\)

Solution:

\(\frac{6y^2}{4x^2}\) × \(\frac{2x^3}{3y^3}\) ÷ \(\frac{5xy}{6ab}\)

\(\frac{6y^2}{4x^2}\) × \(\frac{2x^3}{3y^3}\) × \(\frac{6ab}{5xy}\)

\(\frac{6abx^3y^2}{5x^3y^4}\)

\(\frac{6ab}{5y^{4-2}}\)

\(\frac{6ab}{5y^2}\)

Solution:

Here,  \(\frac{x^2}{a^2}\) × \(\frac{a}{x}\)

 = \(\frac{x^2a}{a^2x}\)

 = \(\frac{x}{a}\) ans.

Solution:

Here, \(\frac{a^3}{x^3}\) ÷ \(\frac{a^2}{x^2}\) 

 = \(\frac{a^3}{x^3}\) × \(\frac{x^2}{a^2}\) 

 = \(\frac{a^3x^2}{x^3a^2}\)

 = \(\frac{a}{x}\) ans.

Solution:

Here, \(\frac{x}{2}\) + \(\frac{x}{6}\)

 = \(\frac{3x + x}{6}\)

 = \(\frac{4x}{6}\)

 = \(\frac{2x}{3}\) ans.

Solution: 

Here, \(\frac{3a}{x}\) - \(\frac{2a}{5x}\) 

 = \(\frac{5×3a - 2a}{5x}\)

 = \(\frac{15a - 2a}{5x}\)

 = \(\frac{13a}{5x}\)

Quiz

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