Chapter 5: Problem 30

Remove the square brackets from $$ \frac{1}{\omega}\left[\frac{s \omega}{s^{2}+\omega^{2}}-s\right] $$

Short Answer

Expert verified

Question: Simplify the given expression: $$\left[\frac{s}{\omega} - \frac{s^2}{s^2 + \omega^2}\right] \div \frac{s}{\omega}$$ Answer: The simplified expression is $$\frac{s\omega - s^3 - s\omega^2}{s^2\omega+\omega^3}$$.

Step by step solution

Find a common denominator

To remove the square brackets, we need to find a common denominator for both terms within the brackets. In this case, the common denominator will be $$(s^2 + \omega^2)$$.

Express both terms with the common denominator

Rewrite both terms having $$(s^2 + \omega^2)$$ as their denominator: $$\frac{s \omega}{s^{2}+\omega^{2}} - \frac{s(s^2 + \omega^2)}{s^2 + \omega^2}$$

Subtract the fractions

Now, subtract the fractions using the common denominator: $$\frac{s\omega - s(s^2 + \omega^2)}{s^2+\omega^2}$$

Distribute the negative sign

Distribute the negative sign in the second term of the numerator: $$\frac{s\omega - s^3 - s\omega^2}{s^2+\omega^2}$$

Simplify the expression

Now, multiply the entire expression by $$\frac{1}{\omega}$$: $$\frac{1}{\omega} \cdot \frac{s\omega - s^3 - s\omega^2}{s^2+\omega^2}$$

Simplify further

Multiplying the numerators and denominators of the fractions, we get: $$\frac{s\omega - s^3 - s\omega^2}{s^2\omega+\omega^3}$$ Now, the square brackets have been removed, and the expression simplified.

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Remove the square brackets from $$ \frac{1}{\omega}\left[\frac{s \omega}{s^{2}+\omega^{2}}-s\right] $$

Short Answer

Step by step solution

Find a common denominator

Express both terms with the common denominator

Subtract the fractions

Distribute the negative sign

Simplify the expression

Simplify further

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