Simplify the complex fraction. [tex]$\frac{\frac{h}{n}+\frac{n}{h}}{\frac{h}{n}-\frac{n}{h}}$[/tex]
Answer (1)
Find a common denominator for the numerator and combine the fractions: n h + h n = nh h 2 + n 2 .
Find a common denominator for the denominator and combine the fractions: n h − h n = nh h 2 − n 2 .
Divide the simplified numerator by the simplified denominator: nh h 2 − n 2 nh h 2 + n 2 = nh h 2 + n 2 × h 2 − n 2 nh .
Simplify the resulting expression by canceling the nh terms: h 2 − n 2 h 2 + n 2 . The final answer is h 2 − n 2 h 2 + n 2 .
Explanation
Problem Analysis We are asked to simplify the complex fraction:
Solution Plan n h − h n n h + h n To simplify this complex fraction, we will first find a common denominator for both the numerator and the denominator. Then, we will divide the simplified numerator by the simplified denominator.
Calculations and Simplification The numerator is n h + h n . The common denominator is nh . So we have: n h + h n = nh h 2 + nh n 2 = nh h 2 + n 2 The denominator is n h − h n . The common denominator is nh . So we have: n h − h n = nh h 2 − nh n 2 = nh h 2 − n 2 Now we divide the simplified numerator by the simplified denominator: nh h 2 − n 2 nh h 2 + n 2 = nh h 2 + n 2 ÷ nh h 2 − n 2 = nh h 2 + n 2 × h 2 − n 2 nh We can cancel the nh terms: h 2 − n 2 h 2 + n 2 So the simplified complex fraction is h 2 − n 2 h 2 + n 2 .
Final Answer The simplified complex fraction is h 2 − n 2 h 2 + n 2 .
Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios. For example, when calculating electrical circuits with parallel resistors, the total resistance can be expressed as a complex fraction. Simplifying such fractions helps in determining the overall resistance efficiently. Also, in physics, when dealing with Doppler effect calculations involving relative velocities, complex fractions often arise. Simplifying these fractions makes it easier to compute the observed frequencies or wavelengths.
