Question

What is the main difference between multiplying/dividing rational expressions and

Answer 1

Step-by-step explanation:

multiplying/dividing rational expressions

The course of action of multiplying and dividing the Rational expressions is same as multiplying and dividing the numeric fractions.

To multiply

• First determine the greatest common factors of the numerator and denominator.

• Then, regrouping the factors to make fractions equal to one.

• Then, multiplying any remaining factors.

For example,

$\frac{5}{14}a^2\:.\:\frac{7}{10a^3}$

Find if there are excluded values - values of a which can generate 0 as a denominator

$10a^3\:=\:0$

$a = 0$

The domain is all a ≠ 0

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}\cdot \frac{d}{e}=\frac{a\:\cdot \:b\:\cdot \:d}{c\:\cdot \:e}$

$\frac{5\cdot \:7a^2}{14\cdot \:10a^3}$

$\mathrm{Refine}$

$\frac{35a^2}{140a^3}$

$\mathrm{Cancel\:the\:common\:factor:}\:35$

$\frac{a^2}{4a^3}$

$\mathrm{Apply\:exponent\:rule}:\quad \frac{x^a}{x^b}=\frac{1}{x^{b-a}}$

As

$\frac{a^2}{a^3}=\frac{1}{a^{3-2}}$

So,

$\frac{1}{4a^{3-2}}$

$\mathrm{Subtract\:the\:numbers:}\:3-2=1$

$\frac{1}{4a}$

To Divide

• First rewriting the division as multiplication by the reciprocal of the denominator

• The remaining steps are then the same as for multiplication.

For example,

$\frac{5x^2}{9}\:\div \frac{15x^3}{27}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$

$\frac{5x^2}{9}\times \frac{27}{15x^3}$

x can not be zero i.e. x ≠ 0

$\frac{5x^2}{9}\times \frac{9}{5x^3}$

$\mathrm{Multiply\:fractions}:\quad \frac{a}{b}\times \frac{c}{d}=\frac{a\:\times \:c}{b\:\times \:d}$

$\frac{5x^2\times \:9}{9\times \:5x^3}$

$\mathrm{Cancel\:the\:common\:factor:}\:5$

$\frac{x^2\times \:9}{9x^3}$

$\mathrm{Cancel\:the\:common\:factor:}\:9$

$\frac{x^2}{x^3}$

$\mathrm{Apply\:exponent\:rule}:\quad \frac{x^a}{x^b}=\frac{1}{x^{b-a}}$

$\frac{x^2}{x^3}=\frac{1}{x^{3-2}}$

$\frac{1}{x^{3-2}}$

$\mathrm{Subtract\:the\:numbers:}\:3-2=1$

$\frac{1}{x}$

Therefore, the main difference between multiplying/dividing rational expressions is during multiplying we

• First determine the greatest common factors of the numerator and denominator

and during dividing we

• First rewrite the division as multiplication by the reciprocal of the denominator

Adding/subtracting rational expressions

If the two rational expressions that we would like to want to add or subtract have the same denominator we just add/subtract the numerators which each other.

For example, when we add two rational expressions

$\frac{x}{x-1}\:+\:\frac{3-x}{x-1}$

$\mathrm{Apply\:rule}\:\frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}$

$\frac{x+3-x}{x-1}$

$\frac{3}{x-1}$

And when we subtract two rational expressions

$\frac{x}{x-1}\:-\:\frac{3-x}{x-1}$

$\mathrm{Apply\:rule}\:\frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}$

$\frac{x-\left(-x+3\right)}{x-1}$

$\frac{2x-3}{x-1}$

Keywords: ration expression, orations

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