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Questions and Answers
How can you rewrite the fractions 2/3 and 3/4 with a common denominator of 12?
2/3 = (2 * 4)/12 and 3/4 = (3 * 3)/12
What is the least common multiple (LCM) of the denominators 3 and 4?
12
What is the sum of the fractions (2 * 4)/12 + (3 * 3)/12?
5/12
How can you simplify a complex fraction, where the numerator and/or denominator contain fractions themselves?
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What is the first step in adding or subtracting algebraic fractions with different denominators?
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What is the first step in simplifying a complex fraction?
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Explain the process of canceling out common factors when simplifying algebraic fractions.
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When adding or subtracting fractions with unlike denominators, what is the first step you must take?
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How do you simplify an algebraic fraction that contains a variable, such as $rac{rac{3}{2}}{x}$?
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What is the simplified form of the complex fraction $rac{1}{x} - rac{3}{y}$, assuming $x$ and $y$ share no common factor?
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Study Notes
Simplifying Algebraic Fractions
Understanding Algebraic Fractions
An algebraic fraction is a fraction whose numerator or denominator contains algebraic expressions containing variable value. There are specific rules to simplify these types of fractions. Here are some ways to simplify algebraic fractions.
Finding the Least Common Denominator
The least common multiple (LCM) of two or more integers is the smallest positive integer that can be divided by all of those integers without remainder. Applying this principle to algebraic fractions, we want to find a common denominator for the fractions we want to combine.
For example, let's simplify the fractions:
2/3 + 3/4
We want to find the LCM of the denominators 3 and 4. In this case, the LCM is 12, as it is the smallest number that can evenly divide both 3 and 4.
Now, we can rewrite the fractions with 12 as their common denominator:
2/3 = (2 * 4)/12
3/4 = (3 * 3)/12
So, now we have:
(2 * 4)/12 + (3 * 3)/12
This reduces to:
2 + 3 = 5
Therefore, the sum of the two fractions is 5/12.
Note: The above example assumes that the variables involved in the fractions are already known or provided.
Simplifying Complex Fractions
A complex fraction is a fraction where the numerator and/or denominator contain fractions themselves. To simplify a complex fraction, start by performing the division within the parentheses, simplifying each fraction separately. Once you have simplified the individual fractions, perform the desired arithmetic operation (addition, subtraction, etc.) on the simplified parts.
Example: Simplify the complex fraction:
\frac{1}{x}-\frac{3}{y}
First, we need to find the common denominator between (x) and (y). Assuming they share no common factor, the common denominator would be (xy).
Now, we can rewrite the complex fraction:
\frac{1}{x}-\frac{3}{y}=\frac{1}{x}\cdot \frac{y}{y}-\frac{3}{y}\cdot \frac{x}{x}= \frac{xy-3x}{xy}
Next, we cancel out any common factors on the top and bottom of the fraction:
\frac{xy-3x}{xy}=\frac{y-3}{x}
So, the simplified complex fraction is (\frac{y-3}{x}).
Canceling Out Common Factors
When simplifying algebraic fractions, if there are common factors shared between the numerator and denominator, you can eliminate them by dividing both the numerator and denominator by the common factor:
\frac{a}{c}=\frac{a \div c}{c \div c}
This process eliminates the common factor from the fraction, making it simpler.
Adding and Subtracting Fractions
Adding or subtracting fractions with unlike denominators involves finding a common denominator and converting the fractions to equivalent fractions with the common denominator. Only after that can the fractions be added or subtracted.
For example, let's add the fractions (\frac{2}{3}) and (\frac{5}{6}):
\frac{2}{3}+\frac{5}{6}=\frac{(2+5)}{(3+6)}=\frac{7}{9}
Simplifying Expressions with Variables
If the algebraic fraction contains variables, like (x), you typically try to isolate the variable in the numerator of the fraction. Here's an example:
\frac{\frac{3}{2}}{x}
In this case, we want to isolate (x) in the numerator, so we can cancel it out:
\frac{\frac{3}{2}}{x}=\frac{3}{2x}
Now, the variable (x) is isolated in the denominator, allowing us to simplify the fraction without worrying about variable manipulation.
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Description
Test your knowledge on simplifying algebraic fractions by understanding how to find the least common denominator, simplify complex and common fractions, cancel out common factors, and add/subtract fractions. Learn how to simplify expressions with variables in algebraic fractions.
