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Chapter 5: Problem 28

Simplify \(\frac{A}{2 z} \frac{1}{s-\omega}-\frac{A}{2 z} \frac{1}{s+\omega}\).

Short Answer

Expert verified
Question: Simplify the given mathematical expression: \(\frac{A}{2z} \frac{1}{s-\omega}-\frac{A}{2z} \frac{1}{s+\omega}\). Answer: \(\frac{A\omega}{z(s-\omega)(s+\omega)}\)

Step by step solution

01

Identify the common factors in both terms of the expression

We have the expression \(\frac{A}{2z} \frac{1}{s-\omega}-\frac{A}{2z} \frac{1}{s+\omega}\). Note that both terms have a common factor of \(\frac{A}{2z}\).
02

Factor out the common term

Factor out \(\frac{A}{2z}\) from both terms. This gives us: \(\frac{A}{2z}\left(\frac{1}{s-\omega}-\frac{1}{s+\omega}\right)\).
03

Simplify the expression inside the parentheses

To simplify the expression inside the parentheses, find a common denominator for the two fractions. The common denominator will be \((s-\omega)(s+\omega)\). Rewrite the fractions with this common denominator: \(\frac{A}{2z}\left(\frac{(s+\omega)-(s-\omega)}{(s-\omega)(s+\omega)}\right)\).
04

Simplify the numerator in the parentheses

Simplify the numerator by combining the terms in the parentheses: \(\frac{A}{2z}\left(\frac{2\omega}{(s-\omega)(s+\omega)}\right)\).
05

Combine the fractions

Combine the fractions by multiplying the numerators and denominators: \(\frac{A(2\omega)}{2z(s-\omega)(s+\omega)}\).
06

Simplify the final expression if possible

We can further simplify the expression by canceling out the 2 in the numerator and denominator: \(\frac{A\omega}{z(s-\omega)(s+\omega)}\). The final simplified expression is: \(\frac{A\omega}{z(s-\omega)(s+\omega)}\).

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