Simplify the complex fraction
$\frac{3-\frac{7}{x}}{x+\frac{2}{3 x}}$
Answer (2)
Multiply the numerator and denominator by 3 x to eliminate the inner fractions.
Distribute and simplify the expression.
Factor the numerator.
The simplified expression is 3 x 2 + 2 3 ( 3 x − 7 ) .
Explanation
Understanding the Problem We are asked to simplify the complex fraction x + 3 x 2 3 − x 7 To do this, we will multiply the numerator and denominator by 3 x to eliminate the inner fractions.
Simplifying the Fraction Multiply the numerator and denominator by 3 x :
x + 3 x 2 3 − x 7 × 3 x 3 x = 3 x ( x + 3 x 2 ) 3 x ( 3 − x 7 ) Distribute 3 x in the numerator and the denominator: 3 x ( x ) + 3 x ( 3 x 2 ) 3 x ( 3 ) − 3 x ( x 7 ) = 3 x 2 + 2 ( 3 x /3 x ) 9 x − 7 ( 3 x / x ) Simplify: 3 x 2 + 2 9 x − 21 Factor out a 3 from the numerator: 3 x 2 + 2 3 ( 3 x − 7 )
Final Answer The simplified complex fraction is: 3 x 2 + 2 3 ( 3 x − 7 )
Examples
Complex fractions appear in various fields, such as physics and engineering, when dealing with impedance in electrical circuits or fluid dynamics. Simplifying these fractions makes calculations easier and provides a clearer understanding of the relationships between different variables. For example, if you are analyzing the flow rate of a fluid through a pipe and the equation involves a complex fraction, simplifying it will help you determine the flow rate more efficiently.
To simplify the complex fraction x + 3 x 2 3 − x 7 , we multiply the numerator and denominator by the common denominator 3 x . This results in the simplified expression 3 x 2 + 2 3 ( 3 x − 7 ) .
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