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Simplify Complex Rational Expressions

Simplify a complex rational expression by writing it as division

Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:

\[\begin{array}{llllll}\frac{\frac{3}{4}}{\frac{5}{8}} & & & & & \frac{\frac{x}{2}}{\frac{xy}{6}}\end{array}\]

In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

Here are a few complex rational expressions:

\[\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}\ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}\ \frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}\]

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

We have already seen this complex rational expression earlier in this chapter.

\[\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}}\]

We noted that fraction bars tell us to divide, so rewrote it as the division problem:

\[(\frac{6{x}^{2}-7x+2}{4x-8})\div (\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}).\]

Example

Try it.

Simplify the complex rational expression by writing it as division: \(\frac{\frac{6}{x-4}}{\frac{3}{{x}^{2}-16}}.\)

Solution

	\(\ \frac{\frac{6}{x-4}}{\frac{3}{{x}^{2}-16}}\)
Rewrite the complex fraction as division.	\(\ \frac{6}{x-4}\div \frac{3}{{x}^{2}-16}\)
Rewrite as the product of first times the reciprocal of the second.	\(\ \frac{6}{x-4}\cdot \frac{{x}^{2}-16}{3}\)
Factor.	\(\ \frac{3\cdot 2}{x-4}\cdot \frac{(x-4)(x+4)}{3}\)
Multiply.	\(\ \frac{3\cdot 2(x-4)(x+4)}{3(x-4)}\)
Remove common factors.	\(\ \frac{3\cdot 2(x-4)(x+4)}{3(x-4)}\)
Simplify.	\(\ 2(x+4)\)

Are there any value(s) of x that should not be allowed? The original complex rational expression had denominators of \(x-4\) and \({x}^{2}-16.\) This expression would be undefined if \(x=4\) or \(x=-4.\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in . We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by \(\frac{\text{LCD}}{\text{LCD}}\) we are multiplying by 1, so the value stays the same.

Example

Try it.

Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.\)

Solution


The LCD of all the fractions in the whole expression is 6.
Clear the fractions by multiplying the numerator and denominator by that LCD.
Distribute.
Simplify.

We will use the same example as in . Decide which method works better for you.

How to Simplify a Complex Rational Expressing using the LCD

Try it.

Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.\)

Solution

Be sure to start by factoring all the denominators so you can find the LCD.

Example

Try it.

Simplify the complex rational expression by using the LCD: \(\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}.\)

Solution


Find the LCD of all fractions in the complex rational expression. The LCD is \({x}^{2}-36=(x+6)(x-6)\).
Multiply the numerator and denominator by the LCD.
Simplify the expression.
Distribute in the denominator.
Simplify.
Simplify.
To simplify the denominator, distribute and combine like terms.
Factor the denominator.
Remove common factors.
Simplify.
Notice that there are no more factors common to the numerator and denominator.

Be sure to factor the denominators first. Proceed carefully as the math can get messy!

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

How to simplify a complex rational expression by writing it as division.
1. Simplify the numerator and denominator.
2. Rewrite the complex rational expression as a division problem.
3. Divide the expressions.
How to simplify a complex rational expression by using the LCD.
1. Find the LCD of all fractions in the complex rational expression.
2. Multiply the numerator and denominator by the LCD.
3. Simplify the expression.

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

Try it.

\(\frac{\frac{2a}{a+4}}{\frac{4{a}^{2}}{{a}^{2}-16}}\)

Solution

\(\frac{a-4}{2a}\)

Try it.

\(\frac{\frac{3b}{b-5}}{\frac{{b}^{2}}{{b}^{2}-25}}\)

Try it.

\(\frac{\frac{5}{{c}^{2}+5c-14}}{\frac{10}{c+7}}\)

Solution

\(\frac{1}{2(c-2)}\)

Try it.

\(\frac{\frac{8}{{d}^{2}+9d+18}}{\frac{12}{d+6}}\)

Try it.

\(\frac{\frac{1}{2}+\frac{5}{6}}{\frac{2}{3}+\frac{7}{9}}\)

Solution

\(\frac{12}{13}\)

Try it.

\(\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}\)

Try it.

\(\frac{\frac{2}{3}-\frac{1}{9}}{\frac{3}{4}+\frac{5}{6}}\)

Solution

\(\frac{20}{57}\)

Try it.

\(\frac{\frac{1}{2}-\frac{1}{6}}{\frac{2}{3}+\frac{3}{4}}\)

Try it.

\(\frac{\frac{n}{m}+\frac{1}{n}}{\frac{1}{n}-\frac{n}{m}}\)

Solution

\(\frac{{n}^{2}+m}{m-{n}^{2}}\)

Try it.

\(\frac{\frac{1}{p}+\frac{p}{q}}{\frac{q}{p}-\frac{1}{q}}\)

Try it.

\(\frac{\frac{1}{r}+\frac{1}{t}}{\frac{1}{{r}^{2}}-\frac{1}{{t}^{2}}}\)

Solution

\(\frac{rt}{t-r}\)

Try it.

\(\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{{v}^{2}}-\frac{1}{{w}^{2}}}\)

Try it.

\(\frac{x-\frac{2x}{x+3}}{\frac{1}{x+3}+\frac{1}{x-3}}\)

Solution

\(\frac{(x+1)(x-3)}{2}\)

Try it.

\(\frac{y-\frac{2y}{y-4}}{\frac{2}{y-4}+\frac{2}{y+4}}\)

Try it.

\(\frac{2-\frac{2}{a+3}}{\frac{1}{a+3}+\frac{a}{2}}\)

Solution

\(\frac{4}{a+1}\)

Try it.

\(\frac{4+\frac{4}{b-5}}{\frac{1}{b-5}+\frac{b}{4}}\)

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

Try it.

\(\frac{\frac{1}{3}+\frac{1}{8}}{\frac{1}{4}+\frac{1}{12}}\)

Solution

\(\frac{11}{8}\)

Try it.

\(\frac{\frac{1}{4}+\frac{1}{9}}{\frac{1}{6}+\frac{1}{12}}\)

Try it.

\(\frac{\frac{5}{6}+\frac{2}{9}}{\frac{7}{18}-\frac{1}{3}}\)

Solution

\(19\)

Try it.

\(\frac{\frac{1}{6}+\frac{4}{15}}{\frac{3}{5}-\frac{1}{2}}\)

Try it.

\(\frac{\frac{c}{d}+\frac{1}{d}}{\frac{1}{d}-\frac{d}{c}}\)

Solution

\(\frac{{c}^{2}+c}{c-{d}^{2}}\)

Try it.

\(\frac{\frac{1}{m}+\frac{m}{n}}{\frac{n}{m}-\frac{1}{n}}\)

Try it.

\(\frac{\frac{1}{p}+\frac{1}{q}}{\frac{1}{{p}^{2}}-\frac{1}{{q}^{2}}}\)

Solution

\(\frac{pq}{q-p}\)

Try it.

\(\frac{\frac{2}{r}+\frac{2}{t}}{\frac{1}{{r}^{2}}-\frac{1}{{t}^{2}}}\)

Try it.

\(\frac{\frac{2}{x+5}}{\frac{3}{x-5}+\frac{1}{{x}^{2}-25}}\)

Solution

\(\frac{2x-10}{3x+16}\)

Try it.

\(\frac{\frac{5}{y-4}}{\frac{3}{y+4}+\frac{2}{{y}^{2}-16}}\)

Try it.

\(\frac{\frac{5}{{z}^{2}-64}+\frac{3}{z+8}}{\frac{1}{z+8}+\frac{2}{z-8}}\)

Solution

\(\frac{3z-19}{3z+8}\)

Try it.

\(\frac{\frac{3}{s+6}+\frac{5}{s-6}}{\frac{1}{{s}^{2}-36}+\frac{4}{s+6}}\)

Try it.

\(\frac{\frac{4}{{a}^{2}-2a-15}}{\frac{1}{a-5}+\frac{2}{a+3}}\)

Solution

\(\frac{4}{3a-7}\)

Try it.

\(\frac{\frac{5}{{b}^{2}-6b-27}}{\frac{3}{b-9}+\frac{1}{b+3}}\)

Try it.

\(\frac{\frac{5}{c+2}-\frac{3}{c+7}}{\frac{5c}{{c}^{2}+9c+14}}\)

Solution

\(\frac{2c+29}{5c}\)

Try it.

\(\frac{\frac{6}{d-4}-\frac{2}{d+7}}{\frac{2d}{{d}^{2}+3d-28}}\)

Try it.

\(\frac{2+\frac{1}{p-3}}{\frac{5}{p-3}}\)

Solution

\(\frac{2p-5}{5}\)

Try it.

\(\frac{\frac{n}{n-2}}{3+\frac{5}{n-2}}\)

Try it.

\(\frac{\frac{m}{m+5}}{4+\frac{1}{m-5}}\)

Solution

\(\frac{m(m-5)}{(4m-19)(m+5)}\)

Try it.

\(\frac{7+\frac{2}{q-2}}{\frac{1}{q+2}}\)

In the following exercises, simplify each complex rational expression using either method.

Try it.

\(\frac{\frac{3}{4}-\frac{2}{7}}{\frac{1}{2}+\frac{5}{14}}\)

Solution

\(\frac{13}{24}\)

Try it.

\(\frac{\frac{v}{w}+\frac{1}{v}}{\frac{1}{v}-\frac{v}{w}}\)

Try it.

\(\frac{\frac{2}{a+4}}{\frac{1}{{a}^{2}-16}}\)

Solution

\(2(a-4)\)

Try it.

\(\frac{\frac{3}{{b}^{2}-3b-40}}{\frac{5}{b+5}-\frac{2}{b-8}}\)

Try it.

\(\frac{\frac{3}{m}+\frac{3}{n}}{\frac{1}{{m}^{2}}-\frac{1}{{n}^{2}}}\)

Solution

\(\frac{3mn}{n-m}\)

Try it.

\(\frac{\frac{2}{r-9}}{\frac{1}{r+9}+\frac{3}{{r}^{2}-81}}\)

Try it.

\(\frac{x-\frac{3x}{x+2}}{\frac{3}{x+2}+\frac{3}{x-2}}\)

Solution

\(\frac{(x-1)(x-2)}{6}\)

Try it.

\(\frac{\frac{y}{y+3}}{2+\frac{1}{y-3}}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Simplify a Complex Rational Expression by Writing it as Division

We have already seen this complex rational expression earlier in this chapter.

\[\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}}\]

We noted that fraction bars tell us to divide, so rewrote it as the division problem

\[(\frac{6{x}^{2}-7x+2}{4x-8})\div (\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6})\]

Then we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify rational expressions. We write it as if we were dividing two fractions.

Example

Try it.

Simplify: \(\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}.\)

Solution

	\(\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}\)
Rewrite the complex fraction as division.	\(\frac{4}{y-3}\div \frac{8}{{y}^{2}-9}\)
Rewrite as the product of first times the reciprocal of the second.	\(\frac{4}{y-3}\cdot \frac{{y}^{2}-9}{8}\)
Multiply.	\(\frac{4({y}^{2}-9)}{8(y-3)}\)
Factor to look for common factors.	\(\frac{4(y-3)(y+3)}{4\cdot 2(y-3)}\)
Remove common factors.	\(\frac{4(y-3)(y+3)}{4\cdot 2(y-3)}\)
Simplify.	\(\frac{y+3}{2}\)

Are there any value(s) of \(y\) that should not be allowed? The simplified rational expression has just a constant in the denominator. But the original complex rational expression had denominators of \(y-3\) and \({y}^{2}-9\). This expression would be undefined if \(y=3\) or \(y=-3\).

Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

Example

Try it.

Simplify: \(\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.\)

Solution


Simplify the numerator and denominator.
Find the LCD and add the fractions in the numerator. Find the LCD and add the fractions in the denominator.
Simplify the numerator and denominator.
Simplify the numerator and denominator, again.
Rewrite the complex rational expression as a division problem.
Multiply the first times by the reciprocal of the second.
Simplify.

How to Simplify a Complex Rational Expression by Writing it as Division

Try it.

Simplify: \(\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.\)

Solution

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Simplify a Complex Rational Expression by Using the LCD

Example

Try it.

Simplify: \(\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.\)

Solution


The LCD of all the fractions in the whole expression is 6.
Clear the fractions by multiplying the numerator and denominator by that LCD.
Distribute.
Simplify.

How to Simplify a Complex Rational Expression by Using the LCD

Try it.

Simplify: \(\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.\)

Solution

Be sure to start by factoring all the denominators so you can find the LCD.

Example

Try it.

Simplify: \(\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}.\)

Solution


Find the LCD of all fractions in the complex rational expression. The LCD is \((x+6)(x-6)\).
Multiply the numerator and denominator by the LCD.
Simplify the expression.
Distribute in the denominator.
Simplify.
Simplify.
To simplify the denominator, distribute and combine like terms.
Remove common factors.
Simplify.
Notice that there are no more factors common to the numerator and denominator.

Example

Try it.

Simplify: \(\frac{\frac{4}{{m}^{2}-7m+12}}{\frac{3}{m-3}-\frac{2}{m-4}}.\)

Solution


Find the LCD of all fractions in the complex rational expression. The LCD is \((m-3)(m-4)\).
Multiply the numerator and denominator by the LCD.
Simplify.
Simplify.
Distribute.
Combine like terms.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

To Simplify a Rational Expression by Writing it as Division
1. Simplify the numerator and denominator.
2. Rewrite the complex rational expression as a division problem.
3. Divide the expressions.
To Simplify a Complex Rational Expression by Using the LCD
1. Find the LCD of all fractions in the complex rational expression.
2. Multiply the numerator and denominator by the LCD.
3. Simplify the expression.

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

Try it.

\(\frac{\frac{2a}{a+4}}{\frac{4{a}^{2}}{{a}^{2}-16}}\)

Solution

\(\frac{a-4}{2a}\)

Try it.

\(\frac{\frac{3b}{b-5}}{\frac{{b}^{2}}{{b}^{2}-25}}\)

Try it.

\(\frac{\frac{5}{{c}^{2}+5c-14}}{\frac{10}{c+7}}\)

Solution

\(\frac{1}{2(c-2)}\)

Try it.

\(\frac{\frac{8}{{d}^{2}+9d+18}}{\frac{12}{d+6}}\)

Try it.

\(\frac{\frac{1}{2}+\frac{5}{6}}{\frac{2}{3}+\frac{7}{9}}\)

Solution

\(\frac{12}{13}\)

Try it.

\(\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}\)

Try it.

\(\frac{\frac{2}{3}-\frac{1}{9}}{\frac{3}{4}+\frac{5}{6}}\)

Solution

\(\frac{20}{57}\)

Try it.

\(\frac{\frac{1}{2}-\frac{1}{6}}{\frac{2}{3}+\frac{3}{4}}\)

Try it.

\(\frac{\frac{n}{m}+\frac{1}{n}}{\frac{1}{n}-\frac{n}{m}}\)

Solution

\(\frac{{n}^{2}+m}{m-{n}^{2}}\)

Try it.

\(\frac{\frac{1}{p}+\frac{p}{q}}{\frac{q}{p}-\frac{1}{q}}\)

Try it.

\(\frac{\frac{1}{r}+\frac{1}{t}}{\frac{1}{{r}^{2}}-\frac{1}{{t}^{2}}}\)

Solution

\(\frac{rt}{t-r}\)

Try it.

\(\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{{v}^{2}}-\frac{1}{{w}^{2}}}\)

Try it.

\(\frac{x-\frac{2x}{x+3}}{\frac{1}{x+3}+\frac{1}{x-3}}\)

Solution

\(\frac{(x+1)(x-3)}{2}\)

Try it.

\(\frac{y-\frac{2y}{y-4}}{\frac{2}{y-4}-\frac{2}{y+4}}\)

Try it.

\(\frac{2-\frac{2}{a+3}}{\frac{1}{a+3}+\frac{a}{2}}\)

Solution

\(\frac{4}{a+1}\)

Try it.

\(\frac{4-\frac{4}{b-5}}{\frac{1}{b-5}+\frac{b}{4}}\)

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Try it.

\(\frac{\frac{1}{3}+\frac{1}{8}}{\frac{1}{4}+\frac{1}{12}}\)

Solution

\(\frac{11}{8}\)

Try it.

\(\frac{\frac{1}{4}+\frac{1}{9}}{\frac{1}{6}+\frac{1}{12}}\)

Try it.

\(\frac{\frac{5}{6}+\frac{2}{9}}{\frac{7}{18}-\frac{1}{3}}\)

Solution

\(19\)

Try it.

\(\frac{\frac{1}{6}+\frac{4}{15}}{\frac{3}{5}-\frac{1}{2}}\)

Try it.

\(\frac{\frac{c}{d}+\frac{1}{d}}{\frac{1}{d}-\frac{d}{c}}\)

Solution

\(\frac{{c}^{2}+c}{c-{d}^{2}}\)

Try it.

\(\frac{\frac{1}{m}+\frac{m}{n}}{\frac{n}{m}-\frac{1}{n}}\)

Try it.

\(\frac{\frac{1}{p}+\frac{1}{q}}{\frac{1}{{p}^{2}}-\frac{1}{{q}^{2}}}\)

Solution

\(\frac{pq}{q-p}\)

Try it.

\(\frac{\frac{2}{r}+\frac{2}{t}}{\frac{1}{{r}^{2}}-\frac{1}{{t}^{2}}}\)

Try it.

\(\frac{\frac{2}{x+5}}{\frac{3}{x-5}+\frac{1}{{x}^{2}-25}}\)

Solution

\(\frac{2x-10}{3x+16}\)

Try it.

\(\frac{\frac{5}{y-4}}{\frac{3}{y+4}+\frac{2}{{y}^{2}-16}}\)

Try it.

\(\frac{\frac{5}{{z}^{2}-64}+\frac{3}{z+8}}{\frac{1}{z+8}+\frac{2}{z-8}}\)

Solution

\(\frac{3z-19}{3z+8}\)

Try it.

\(\frac{\frac{3}{s+6}+\frac{5}{s-6}}{\frac{1}{{s}^{2}-36}+\frac{4}{s+6}}\)

Try it.

\(\frac{\frac{4}{{a}^{2}-2a-15}}{\frac{1}{a-5}+\frac{2}{a+3}}\)

Solution

\(\frac{4}{3a-7}\)

Try it.

\(\frac{\frac{5}{{b}^{2}-6b-27}}{\frac{3}{b-9}+\frac{1}{b+3}}\)

Try it.

\(\frac{\frac{5}{c+2}-\frac{3}{c+7}}{\frac{5c}{{c}^{2}+9c+14}}\)

Solution

\(\frac{2c+29}{5c}\)

Try it.

\(\frac{\frac{6}{d-4}-\frac{2}{d+7}}{\frac{2d}{{d}^{2}+3d-28}}\)

Try it.

\(\frac{2+\frac{1}{p-3}}{\frac{5}{p-3}}\)

Solution

\(\frac{(2p-5)}{5}\)

Try it.

\(\frac{\frac{n}{n-2}}{3+\frac{5}{n-2}}\)

Try it.

\(\frac{\frac{m}{m+5}}{4+\frac{1}{m-5}}\)

Solution

\(\frac{m(m-5)}{4{m}^{2}+m-95}\)

Try it.

\(\frac{7+\frac{2}{q-2}}{\frac{1}{q+2}}\)

Simplify

In the following exercises, use either method.

Try it.

\(\frac{\frac{3}{4}-\frac{2}{7}}{\frac{1}{2}+\frac{5}{14}}\)

Solution

\(\frac{13}{24}\)

Try it.

\(\frac{\frac{v}{w}+\frac{1}{v}}{\frac{1}{v}-\frac{v}{w}}\)

Try it.

\(\frac{\frac{2}{a+4}}{\frac{1}{{a}^{2}-16}}\)

Solution

\(2(a-4)\)

Try it.

\(\frac{\frac{3}{{b}^{2}-3b-40}}{\frac{5}{b+5}-\frac{2}{b-8}}\)

Try it.

\(\frac{\frac{3}{m}+\frac{3}{n}}{\frac{1}{{m}^{2}}-\frac{1}{{n}^{2}}}\)

Solution

\(\frac{3mn}{n-m}\)

Try it.

\(\frac{\frac{2}{r-9}}{\frac{1}{r+9}+\frac{3}{{r}^{2}-81}}\)

Try it.

\(\frac{x-\frac{3x}{x+2}}{\frac{3}{x+2}+\frac{3}{x-2}}\)

Solution

\(\frac{(x-1)(x-2)}{6}\)

Try it.

\(\frac{\frac{y}{y+3}}{2+\frac{1}{y-3}}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: \(\frac{\frac{3}{5}}{\frac{9}{10}}.\)
If you missed this problem, review .

উত্তর প্রকাশ করুন

\(\frac{2}{3}\)
Simplify: \(\frac{1-\frac{1}{3}}{{4}^{2}+4\cdot 5}.\)
If you missed this problem, review .

উত্তর প্রকাশ করুন

\(\frac{1}{54}\)
Solve: \(\frac{1}{2x}+\frac{1}{4}=\frac{1}{8}.\)
If you missed this problem, review .

উত্তর প্রকাশ করুন

\(x=-4\)

Simplify the complex rational expression by writing it as division: \(\frac{\frac{6}{x-4}}{\frac{3}{{x}^{2}-16}}.\)

উত্তর প্রকাশ করুন

	\(\ \frac{\frac{6}{x-4}}{\frac{3}{{x}^{2}-16}}\)
Rewrite the complex fraction as division.	\(\ \frac{6}{x-4}\div \frac{3}{{x}^{2}-16}\)
Rewrite as the product of first times the reciprocal of the second.	\(\ \frac{6}{x-4}\cdot \frac{{x}^{2}-16}{3}\)
Factor.	\(\ \frac{3\cdot 2}{x-4}\cdot \frac{(x-4)(x+4)}{3}\)
Multiply.	\(\ \frac{3\cdot 2(x-4)(x+4)}{3(x-4)}\)
Remove common factors.	\(\ \frac{3\cdot 2(x-4)(x+4)}{3(x-4)}\)
Simplify.	\(\ 2(x+4)\)

Simplify the complex rational expression by writing it as division: \(\frac{\frac{2}{{x}^{2}-1}}{\frac{3}{x+1}}.\)

উত্তর প্রকাশ করুন

\(\frac{2}{3(x-1)}\)
Simplify the complex rational expression by writing it as division: \(\frac{\frac{1}{{x}^{2}-7x+12}}{\frac{2}{x-4}}.\)

উত্তর প্রকাশ করুন

\(\frac{1}{2(x-3)}\)

Simplify the complex rational expression by writing it as division: \(\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.\)

উত্তর প্রকাশ করুন


Simplify the numerator and denominator. Find the LCD and add the fractions in the numerator. Find the LCD and subtract the fractions in the denominator.
Simplify the numerator and denominator.
Rewrite the complex rational expression as a division problem.
Multiply the first by the reciprocal of the second.
Simplify.	3

Simplify the complex rational expression by writing it as division: \(\frac{\frac{1}{2}+\frac{2}{3}}{\frac{5}{6}+\frac{1}{12}}.\)

উত্তর প্রকাশ করুন

\(\frac{14}{11}\)
Simplify the complex rational expression by writing it as division: \(\frac{\frac{3}{4}-\frac{1}{3}}{\frac{1}{8}+\frac{5}{6}}.\)

উত্তর প্রকাশ করুন

\(\frac{10}{23}\)
Simplify the complex rational expression by writing it as division: \(\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.\)

উত্তর প্রকাশ করুন
Simplify the complex rational expression by writing it as division: \(\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}.\)

উত্তর প্রকাশ করুন

\(\frac{y+x}{y-x}\)
Simplify the complex rational expression by writing it as division: \(\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{{a}^{2}}-\frac{1}{{b}^{2}}}\).

উত্তর প্রকাশ করুন

\(\frac{ab}{b-a}\)

Simplify the complex rational expression by writing it as division: \(\frac{n-\frac{4n}{n+5}}{\frac{1}{n+5}+\frac{1}{n-5}}.\)

উত্তর প্রকাশ করুন


Simplify the numerator and denominator. Find common denominators for the numerator and denominator.
Simplify the numerators.
Subtract the rational expressions in the numerator and add in the denominator.
Simplify. (We now have one rational expression over one rational expression.)
Rewrite as fraction division.
Multiply the first times the reciprocal of the second.
Factor any expressions if possible.
Remove common factors.
Simplify.

Simplify the complex rational expression by writing it as division: \(\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}.\)

উত্তর প্রকাশ করুন

\(\frac{b(b+2)(b-5)}{3b-5}\)
Simplify the complex rational expression by writing it as division: \(\frac{1-\frac{3}{c+4}}{\frac{1}{c+4}+\frac{c}{3}}.\)

উত্তর প্রকাশ করুন

\(\frac{3}{c+3}\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.\)

উত্তর প্রকাশ করুন

The LCD of all the fractions in the whole expression is 6.
Clear the fractions by multiplying the numerator and
denominator by that LCD.
Distribute.
Simplify.
Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{2}+\frac{1}{5}}{\frac{1}{10}+\frac{1}{5}}.\)

উত্তর প্রকাশ করুন

\(\frac{7}{3}\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{4}+\frac{3}{8}}{\frac{1}{2}-\frac{5}{16}}.\)

উত্তর প্রকাশ করুন

\(\frac{10}{3}\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{a}+\frac{1}{b}}{\frac{a}{b}+\frac{b}{a}}.\)

উত্তর প্রকাশ করুন

\(\frac{b+a}{{a}^{2}+{b}^{2}}\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{1}{{x}^{2}}-\frac{1}{{y}^{2}}}{\frac{1}{x}+\frac{1}{y}}.\)

উত্তর প্রকাশ করুন

\(\frac{y-x}{xy}\)

Simplify the complex rational expression by using the LCD: \(\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}.\)

উত্তর প্রকাশ করুন


Find the LCD of all fractions in the complex rational expression. The LCD is \({x}^{2}-36=(x+6)(x-6)\).
Multiply the numerator and denominator by the LCD.
Simplify the expression.
Distribute in the denominator.
Simplify.
Simplify.
To simplify the denominator, distribute and combine like terms.
Factor the denominator.
Remove common factors.
Simplify.
Notice that there are no more factors common to the numerator and denominator.

Simplify the complex rational expression by using the LCD: \(\frac{\frac{3}{x+2}}{\frac{5}{x-2}-\frac{3}{{x}^{2}-4}}.\)

উত্তর প্রকাশ করুন

\(\frac{3(x-2)}{5x+7}\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{2}{x-7}-\frac{1}{x+7}}{\frac{6}{x+7}-\frac{1}{{x}^{2}-49}}.\)

উত্তর প্রকাশ করুন

\(\frac{x+21}{6x-43}\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{4}{{m}^{2}-7m+12}}{\frac{3}{m-3}-\frac{2}{m-4}}.\)

উত্তর প্রকাশ করুন

Find the LCD of all fractions in the
complex rational expression.
The LCD is \((m-3)(m-4).\)
Multiply the numerator and
denominator by the LCD.
Simplify.
Simplify.
Distribute.
Combine like terms.
Simplify the complex rational expression by using the LCD: \(\frac{\frac{3}{{x}^{2}+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}}.\)

উত্তর প্রকাশ করুন

\(\frac{3}{5x+22}\)
Simplify the complex rational expression by using the LCD: \(\frac{\frac{4y}{y+5}+\frac{2}{y+6}}{\frac{3y}{{y}^{2}+11y+30}}.\)

উত্তর প্রকাশ করুন

\(\frac{2(2{y}^{2}+13y+5)}{3y}\)

Simplify the complex rational expression by using the LCD: \(\frac{\frac{y}{y+1}}{1+\frac{1}{y-1}}.\)

উত্তর প্রকাশ করুন


Find the LCD of all fractions in the complex rational expression.
The LCD is \((y+1)(y-1).\)
Multiply the numerator and denominator by the LCD.
Distribute in the denominator and simplify.
Simplify.
Simplify the denominator and leave the numerator factored.

Factor the denominator and remove factors common with the numerator.
Simplify.

Simplify the complex rational expression by using the LCD: \(\frac{\frac{x}{x+3}}{1+\frac{1}{x+3}}.\)

উত্তর প্রকাশ করুন

\(\frac{x}{x+4}\)
Simplify the complex rational expression by using the LCD: \(\frac{1+\frac{1}{x-1}}{\frac{3}{x+1}}.\)

উত্তর প্রকাশ করুন

\(\frac{x(x+1)}{3(x-1)}\)
\(\frac{\frac{2a}{a+4}}{\frac{4{a}^{2}}{{a}^{2}-16}}\)

উত্তর প্রকাশ করুন

\(\frac{a-4}{2a}\)
\(\frac{\frac{3b}{b-5}}{\frac{{b}^{2}}{{b}^{2}-25}}\)
\(\frac{\frac{5}{{c}^{2}+5c-14}}{\frac{10}{c+7}}\)

উত্তর প্রকাশ করুন

\(\frac{1}{2(c-2)}\)
\(\frac{\frac{8}{{d}^{2}+9d+18}}{\frac{12}{d+6}}\)
\(\frac{\frac{1}{2}+\frac{5}{6}}{\frac{2}{3}+\frac{7}{9}}\)

উত্তর প্রকাশ করুন

\(\frac{12}{13}\)
\(\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}\)
\(\frac{\frac{2}{3}-\frac{1}{9}}{\frac{3}{4}+\frac{5}{6}}\)

উত্তর প্রকাশ করুন

\(\frac{20}{57}\)
\(\frac{\frac{1}{2}-\frac{1}{6}}{\frac{2}{3}+\frac{3}{4}}\)
\(\frac{\frac{n}{m}+\frac{1}{n}}{\frac{1}{n}-\frac{n}{m}}\)

উত্তর প্রকাশ করুন

\(\frac{{n}^{2}+m}{m-{n}^{2}}\)
\(\frac{\frac{1}{p}+\frac{p}{q}}{\frac{q}{p}-\frac{1}{q}}\)

Symbols used here

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Simplify Complex Rational Expressions

Simplify a complex rational expression by writing it as division
Simplify a complex rational expression by using the LCD
Simplify the numerator and denominator.
Rewrite the complex rational expression as a division problem.
Divide the expressions.
Find the LCD of all fractions in the complex rational expression.
Multiply the numerator and denominator by the LCD.
Simplify the expression.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

নিজের চেষ্টা করো

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

আরও Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value


Find the LCD of all fractions in the complex rational expression.
The LCD is \((m-3)(m-4).\)
Multiply the numerator and denominator by the LCD.
Simplify.
Simplify.
Distribute.
Combine like terms.