Simplify the complex fraction. Use either method.
$\frac{\frac{1}{y}-\frac{1}{x}}{\frac{1}{y^2}+\frac{1}{x^2}}$
Answer (2)
Find a common denominator for the numerator and combine the fractions: y 1 − x 1 = x y x − y .
Find a common denominator for the denominator and combine the fractions: y 2 1 + x 2 1 = x 2 y 2 x 2 + y 2 .
Divide the simplified numerator by the simplified denominator: x 2 y 2 x 2 + y 2 x y x − y = x y x − y × x 2 + y 2 x 2 y 2 .
Simplify the resulting expression by cancelling common factors: x 2 + y 2 ( x − y ) x y .
The simplified complex fraction is x 2 + y 2 x y ( x − y ) .
Explanation
Problem Setup We are asked to simplify the complex fraction:
Strategy y 2 1 + x 2 1 y 1 − x 1 We will simplify the numerator and the denominator separately, and then divide.
Simplifying the Numerator The numerator is y 1 − x 1 . The common denominator is x y , so we have: y 1 − x 1 = x y x − x y y = x y x − y
Simplifying the Denominator The denominator is y 2 1 + x 2 1 . The common denominator is x 2 y 2 , so we have: y 2 1 + x 2 1 = x 2 y 2 x 2 + x 2 y 2 y 2 = x 2 y 2 x 2 + y 2
Dividing Numerator by Denominator Now we divide the simplified numerator by the simplified denominator: x 2 y 2 x 2 + y 2 x y x − y = x y x − y ÷ x 2 y 2 x 2 + y 2 = x y x − y × x 2 + y 2 x 2 y 2 = x y ( x 2 + y 2 ) ( x − y ) x 2 y 2
Simplifying the Expression We can cancel a factor of x y from the numerator and denominator: x y ( x 2 + y 2 ) ( x − y ) x 2 y 2 = x 2 + y 2 ( x − y ) x y
Final Result Thus, the simplified complex fraction is: x 2 + y 2 x y ( x − y )
Examples
Complex fractions might seem abstract, but they appear in various fields. For example, in electrical engineering, when analyzing parallel circuits, you often encounter complex fractions involving impedances. Simplifying these fractions helps in determining the overall equivalent impedance, which is crucial for circuit design and analysis. Similarly, in physics, complex fractions can arise in calculations involving wave phenomena or optics, where understanding the simplified form aids in predicting wave behavior or designing optical systems.
To simplify the complex fraction y 2 1 + x 2 1 y 1 − x 1 , we first combine the numerator and denominator separately using common denominators. The final result is x 2 + y 2 x y ( x − y ) .
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