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Simplifying Rational Expressions

A rational number is a number of the form $\frac{p}{q}$ where p and q are integers. When the denominator and numerator in $\frac{p}{q}$ both are algebraic expression then the term is known as a rational expression. A rational expression is an expression of the form $\frac{P(x)}{Q(x)}$ over the set of real numbers and Q(x) $\neq$ 0. P(x) and Q(x) represent two polynomials. If there are some common terms in the numerator and the denominator, then it can be taken out in order to simplify the expression. Simplifying rational expression means reducing the rational expression into it's simplest form.

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How to Simplify Rational Expressions?

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Let us try and understand using examples:

Rational algebraic expressions $\frac{P(x)}{Q(x)}$ can be reduced to it's lowest term by dividing both numerator P(x) and denominator Q(x) by the G.C.D. of P(x) and Q(x).

Let us simplify the rational algebraic expression $\frac{5x + 20}{8x + 32}$

$\frac{5x + 20}{8x + 32}$ = $\frac{5(x + 4)}{8(x + 4)}$

Cancel (x + 4) term from both numerator and denominator,

$\frac{5x + 20}{8x + 32}$ = $\frac{5}{8}$


Simplifying Complex Rational Expressions

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A complex fraction is a fraction that has a fraction in its numerator or denominator, or both. To solve complex fraction, invert the denominator fraction and multiply. Let us see with the help of an example how to solve complex fractions.

Let us simplify a complex rational expression $\frac{\frac{1}{a^2} - \frac{1}{b^2}}{\frac{1}{a^2} + \frac{1}{b^2} - \frac{2}{ab}}$

Given expression is $\frac{\frac{1}{a^2} - \frac{1}{b^2}}{\frac{1}{a^2} + \frac{1}{b^2} - \frac{2}{ab}}$

Step 1:
Solve numerator

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

Step 2:
Solve denominator

$\frac{1}{a^2} + \frac{1}{b^2} - \frac{2}{ab}$ = $(\frac{1}{a} - \frac{1}{b})^2$

= $\frac{(b-a)^2}{(ab)^2}$


Now,
$\frac{\frac{b^2 - a^2}{a^2b^2}}{\frac{(b-a)^2}{(ab)^2}}$

$\frac{b^2 - a^2}{a^2b^2}$ $\times$ $\frac{(ab)^2}{(b - a)^2}$
(Invert the denominator)

$\frac{(b - a)(b + a)}{a^2b^2}$ $\times$ $\frac{a^2b^2}{(b - a)^2}$

Cancel (b - a) and $a^2b^2$ from the numerator and denominator.

$\frac{b + a}{b - a}$


Simplifying Rational Expressions with Exponents

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To simplify rational expressions with exponents and radicals, one method is to begin by factoring out the common factor with the smaller exponent:

Let us solve $\frac{2x^2 - 9x - 5}{x^3 + x^2 - 30x}$

Given expression is $\frac{2x^2 - 9x - 5}{x^3 + x^2 - 30x}$

Factors of numerator, 2$x^2$ - 9x - 5 = (2x + 1)(x - 5)

Factors of denominator, $x^2$ + x - 30 = (x - 5)(x + 6)

Now,

$\frac{2x^2 - 9x - 5}{x^3 + x^2 - 30x}$ = $\frac{(2x + 1)(x - 5)}{x (x - 5)(x + 6)}$

By cancelling common terms from the numerator and denominator, we get

= $\frac{2x + 1}{x (x + 6)}$

Examples

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Given below are some of the examples in simplifying rational expressions.

Solved Examples

Question 1: Simplify $\frac{5x + 20}{6x + 24}$
Solution:
$\frac{5x + 20}{6x + 24}$ = $\frac{5(x+4)}{6(x + 4)}$

Cancel (x + 4) term from both numerator and denominator,

$\frac{5x + 20}{6x + 24}$ = $\frac{5}{6}$



Question 2: Simplify the expression $\frac{4x + 16}{4x + 24}$
Solution:

$\frac{4x + 16}{4x + 24}$ = $\frac{4(x+4)}{4(x + 6)}$

Cancel the number 4 from both numerator and denominator,

$\frac{4x + 16}{4x + 24}$ = $\frac{x + 4}{x + 6}$



Question 3: Simplify rational algebraic expression $\frac{(x^2-x-6)}{(x^2+5x+6)}$
Solution:

Given expression is $\frac{(x^2-x-6)}{(x^2+5x+6)}$

Factors of numerator, $x^2$ - x - 6 = (x - 3)(x + 2)

Factors of denominator, $x^2$ + 5x + 6 = (x + 2)(x + 3)

Now,

$\frac{(x^2 - x - 6)}{(x^2 + 5x + 6)}$ = $\frac{(x - 3)(x + 2)}{(x + 2)(x + 3)}$

Cancel (x + 2) term from both numerator and denominator,

$\frac{(x^2 - x - 6)}{(x^2 + 5x + 6)}$ = $\frac{(x -3)}{(x + 3)}$



Question 4: Simplify $\frac{(x^2+7x+10)}{(x^2-4)}$
Solution:

Factors of numerator, $x^2 + 7x + 10$ = (x + 2)(x + 5)

Factors of denominator, $x^2 - 4$ = (x - 2)(x + 2)

$\frac{(x^2 + 7x + 10)}{(x^2 - 4)}$ = $\frac{(x + 2)(x + 5)}{(x - 2)(x + 2)}$

Term (x + 2) is common term, cancel (x + 2) term from both numerator and denominator.

= $\frac{x + 5}{x - 2}$



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