How To
Given a complex rational expression, simplify it.
Combine the expressions in the numerator into a single rational expression by adding or subtracting.
Combine the expressions in the denominator into a single rational expression by adding or subtracting.
Rewrite as the numerator divided by the denominator.
Rewrite as multiplication.
Multiply.
Simplify.
Simplifying complex rational expressions
Simplify:
$\frac{y+\frac{1}{x}}{\frac{x}{y}}$ .
Begin by combining the expressions in the numerator into one expression.
$$\begin{array}{cc}y\cdot \frac{x}{x}+\frac{1}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Multiplyby}\frac{x}{x}\text{togetLCDasdenominator}.\hfill \\ \frac{xy}{x}+\frac{1}{x}\hfill & \\ \frac{xy+1}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Addnumerators}.\hfill \end{array}$$
Now the numerator is a single rational expression and the denominator is a single rational expression.
$\frac{\frac{xy+1}{x}}{\frac{x}{y}}$
We can rewrite this as division, and then multiplication.
$$\begin{array}{cc}\frac{xy+1}{x}\xf7\frac{x}{y}\hfill & \\ \frac{xy+1}{x}\cdot \frac{y}{x}\hfill & \phantom{\rule{2em}{0ex}}\text{Rewriteasmultiplication}\text{.}\hfill \\ \frac{y(xy+1)}{{x}^{2}}\hfill & \phantom{\rule{2em}{0ex}}\text{Multiply}\text{.}\hfill \end{array}$$
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Q&A
Can a complex rational expression always be simplified?
Yes. We can always rewrite a complex rational expression as a simplified rational expression.
Key concepts
Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See
[link] .
We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See
[link] .
To divide rational expressions, multiply by the reciprocal of the second expression. See
[link] .
Adding or subtracting rational expressions requires finding a common denominator. See
[link] and
[link] .
Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See
[link] .
Section exercises
Verbal
How can you use factoring to simplify rational expressions?
You can factor the numerator and denominator to see if any of the terms can cancel one another out.
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Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.
True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.
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Algebraic
For the following exercises, simplify the rational expressions.
For the following exercises, multiply the rational expressions and express the product in simplest form.
Questions & Answers
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Miranda
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jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Jeffrey
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Miranda
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Miranda
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Steve
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Steve
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Jeffrey
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Jeffrey
what is the solution of the given equation?
please where is the equation
Miranda
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is the identity of 1-cos²5x equal to?
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SORIE
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SORIE
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Yaona
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SORIE
what is the function of sine with respect of cosine , graphically
sinx sin2x is linearly dependent
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function
Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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