Subject: Compulsory Maths

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**

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,

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

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

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

- 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.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

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

Solution:

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

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

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

= \(\frac{2xy}{3ab}\)

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

Solution:

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

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

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

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

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

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

Solution:

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

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

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

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

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

Simplify quickly: \(\frac{x^2}{a^2}\) × \(\frac{a}{x}\)

Solution:

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

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

= \(\frac{x}{a}\) ans.

Divide: \(\frac{a^3}{x^3}\) ÷ \(\frac{a^2}{x^2}\)

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.

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

Solution:

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

= \(\frac{3x + x}{6}\)

= \(\frac{4x}{6}\)

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

Subtract: \(\frac{3a}{x}\) - \(\frac{2a}{5x}\)

Solution:

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

= \(\frac{5×3a - 2a}{5x}\)

= \(\frac{15a - 2a}{5x}\)

= \(\frac{13a}{5x}\)

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