Question

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

Short answer

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

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 $\frac{A}{2z}$.

Factor out the common term

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

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: $\frac{A}{2z}\left(\frac{(s+\omega )-(s-\omega )}{(s-\omega )(s+\omega )}\right)$.

Simplify the numerator in the parentheses

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

Combine the fractions

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

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: $\frac{A\omega }{z(s-\omega )(s+\omega )}$.

Key concepts

Factorization

Factorization is like finding building blocks that are common in terms of an expression. In our example, the expression $\frac{A}{2z}\frac{1}{s-\omega }-\frac{A}{2z}\frac{1}{s+\omega }$ has a common factor of $\frac{A}{2z}$ in both terms. Recognizing this allows us to "factor it out" which simplifies the expression to: $\frac{A}{2z}(\frac{1}{s-\omega }-\frac{1}{s+\omega })$. By removing the common factor, simplifying further operations becomes more manageable.

Common Denominator

When dealing with fractions in an expression, finding a common denominator is crucial to simplifying. In our case, we need to subtract $\frac{1}{s-\omega }$ and $\frac{1}{s+\omega }$. The least common denominator here is $(s-\omega )(s+\omega )$, allowing us to combine the fractions. We rewrite the expression inside the parenthesis as:

• $\frac{(s+\omega )}{(s-\omega )(s+\omega )}-\frac{(s-\omega )}{(s-\omega )(s+\omega )}$

This step is essential for simplifying the expression further.

Fractions

Fractions represent parts of a whole and can sometimes get tricky when combining or simplifying. In our expression, we deal with subtracting fractions with like terms. Once rewritten with the common denominator found earlier, it's crucial to focus on the numerators:

• $\frac{(s+\omega )-(s-\omega )}{(s-\omega )(s+\omega )}$

This demonstrates a subtraction of numerators while maintaining the common denominator, helping further simplify the overall expression.

Numerator and Denominator Simplification

To simplify the fraction, we focus on combining the numerators properly. With the expression
$\frac{(s+\omega )-(s-\omega )}{(s-\omega )(s+\omega )}$,
we can simplify the numerator to $2\omega$. This makes the fraction $\frac{2\omega }{(s-\omega )(s+\omega )}$.
Combine this perspective with the common factor from the overall expression: $\frac{A}{2z}$.

Now, you can simplify further to:

• $\frac{A(2\omega )}{2z(s-\omega )(s+\omega )}$
• Which reduces to $\frac{A\omega }{z(s-\omega )(s+\omega )}$ by canceling out the "2".

Recognizing simplifiable elements in the numerator and denominator is key in reaching the most reduced form.