**How To Solve Complex Fractions With Variables** – This is Section 7.4 (Section 1.0) “Complex Rational Expressions” in Beginning Algebra. For more information on this (including licensing), click here.

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## How To Solve Complex Fractions With Variables

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### Question Video: Simplifying An Algebraic Expression Involving Negative And Fractional Exponents

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## Nested Fractions (video)

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Complex Fraction A fraction whose numerator or denominator consists of one or more fractions. the numerator or denominator is a fraction consisting of one or more fractions. For example,

Simplifying such a fraction requires us to find an equivalent fraction with a whole number and denominator. One way to do this is by partitioning. Remember that dividing a fraction is multiplying the opposite of the denominator.

#### Question Video: Finding The Modulus Of Complex Numbers Using Conjugates

Another way to simplify this complex fraction is to multiply both the numerator and denominator by the LCD of all given fractions. LCD = 4 in this case.

Complex Rational Expression A rational expression in which the numerator or denominator consists of one or more rational expressions. is defined as a rational expression containing one or more rational expressions in the numerator, denominator, or both. For example,

We simplify complex rational expressions by finding equivalent fractions where the numerator and denominator are polynomials. As mentioned above, there are two ways to simplify complex rational expressions, and we will outline the steps in both ways. For clarity, assume that the variable expressions used as denominators are nonzero.

## Solving Basic Equations & Inequalities (one Variable, Linear)

We begin by discussing the simplification of complex rational expressions using division. Before multiplying by the reciprocal of the denominator, we must simplify the numerator and denominator separately. The goal is to first get unit algebraic fractions in the numerator and denominator. The following example shows the steps to simplify complex algebraic fractions.

Step 1: Simplify the numerator and denominator. The goal is to divide one algebraic fraction by another unit algebraic fraction. In this example, before adding or subtracting, find terms in both the numerator and denominator that have a common denominator.

Solution: The LCD of the rational expression in the numerator and denominator is x2. Multiply by the appropriate factor, make the equivalent terms the denominator, and then subtract.

### Adding And Subtracting Algebraic Fractions

Now we have one rational expression divided by another rational expression. Then multiply the numerator by the opposite of the denominator, then multiply and subtract.

Another way to simplify complex rational expressions is to clear the fraction by multiplying the special form expression by 1. In this method, the numerator and denominator of all given fractions are multiplied by the Least Common Divisor (LCD).

Step 1: Determine the LCD of all fractions in the numerator and denominator. In this case, the divisors of the given fraction are 2, x, 4, and x2. So the LCD is 4×2.

### Simplifying Complex Fractions (video)

Step 2: Multiply the numerator and denominator by the LCD. This step should clear the fractions in the numerator and denominator.

This leaves us with polynomial algebraic fractions in the numerator and denominator.

This was the same problem we started this section with, and the results here are the same. It’s worth taking the time to compare the steps involved in using the two methods on the same problem.

## Openalgebra.com: Complex Rational Expressions

Solution: Taking all the divisors into account, the LCD is x2. So multiply the numerator and denominator by x2.

At this point we have a rational expression that can be simplified by factoring and then canceling the common factors.

It is important to note that multiplying the numerator and denominator by the same non-zero factor is equivalent to multiplying by 1 and does not change the problem. Since x2x2=1, the numerator and denominator from the previous example can be multiplied by x2 to obtain an equivalent expression.

### Simplifying Fractions With Negative Exponents

Solution: The LCM of all divisors is (x+1)(x−3). Begin by multiplying the numerator and denominator by these factors.

61. Choose a problem from this set of exercises, write it clearly on paper and explain each step in words. Scan your page and post it to the discussion board.

62. Explain why we must simplify the numerator and denominator to a single algebraic fraction before multiplying by the reciprocal of the denominator.

### Effects Of Waxes And The Related Chemicals On Asphaltene Aggregation And Deposition Phenomena: Experimental And Modeling Studies

63. This section presents two methods of simplifying complex rational expressions. Which of the two methods do you think is more effective and why? We use cookies to make them awesome. By using our site, you agree to our cookie policy. Cookie settings

This article was co-authored by David Jia. David Jia is an academic tutor and founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades on a variety of subjects, as well as providing college admissions counseling, SAT, ACT, ISEE and other test preparation. After scoring an 800 in math and a 690 in English, David received a Dickinson Scholarship from the University of Miami and graduated with a Bachelor of Business Administration. Additionally, David has worked as an online video instructor for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

A complex fraction is a fraction that contains a fraction in the numerator, denominator, or both. For this reason, complex fractions are sometimes called “ordered fractions.” Simplifying a complex fraction is a process that goes from easy to difficult depending on how many terms are in the numerator and denominator, whether any of the terms are variables, and if so, the complexity of the variable terms. See step 1 below to get started!

### Free Online Math Manipulatives For At Home Learning

This article was co-authored by David Jia. David Jia is an academic tutor and founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades on a variety of subjects, as well as providing college admissions counseling, SAT, ACT, ISEE and other test preparation. After scoring an 800 in math and a 690 in English, David received a Dickinson Scholarship from the University of Miami and graduated with a Bachelor of Business Administration. Additionally, David has worked as an online video instructor for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 263,056 times.

To simplify complex fractions, start by finding the opposite of the denominator, which can be done simply by reversing the fraction. This new fraction is then multiplied by the numerator. You should now have a simple fraction. Finally, find the greatest common factor between the numerator and denominator and simplify the new fraction by dividing both fractions by that number. If you want to learn how to simplify fractions with variables, keep reading! We use cookies to make them awesome. By using our site, you agree to our cookie policy. Cookie settings

This article was co-authored by Staff. Our team of trained editors and researchers ensure that articles are accurate and detailed. The content management team closely monitors the work of the editorial team to ensure that each article is supported by reliable research and meets high quality standards.

## Improving The Interpretation Of Small Molecule Diffusion Coefficients

A fraction that includes a fraction in the numerator and denominator is called a complex fraction. These types of expressions can be intimidating, especially when they are algebraic expressions with variables. Simplifying them becomes easier when you remember that the fraction string is the same thing as the division symbol. To simplify complex fractions, first convert them to division problems. Then divide any fraction as if it were dividing by a fraction. Don’t forget to take the opposite side of the second fraction and multiply it. When working with variables, it is important to remember certain algebraic rules to simplify expressions.

This article was co-authored by Staff. Articles are validated by our team of trained editors and researchers