Bobbie Comstock

2022-01-12

Solve the given problem.
In analyzing the tuning of an electronic circuit, the expression
${\left[\omega {\omega }_{0}^{-1}-{\omega }_{0}{\omega }^{-1}\right]}^{2}$
is used. Expand and simplify this expression.

jean2098

Step 1
To simplify the following expression.
${\left[\omega {\omega }_{0}^{-1}-{\omega }_{0}{\omega }^{-1}\right]}^{2}$
Evaluate ${\omega }_{0}^{-1}$ and ${\omega }^{-1}$ using this exponent rule ${a}^{n}=\frac{1}{{a}^{n}}$ as follows:
$={\left[\frac{\omega }{{\omega }_{0}}-\frac{{\omega }_{0}}{\omega }\right]}^{2}$
To subtract these fractions, consider the LCM: $\left(\omega {\omega }_{0}\right)$ as common denominator
Multiply the first fraction by $\omega$ and the second fraction by $\omega$ as follows:
$={\left[\frac{\omega {\omega }_{0}}{\omega {\omega }_{0}}-\frac{\omega {\omega }_{0}}{\omega {\omega }_{0}}\right]}^{2}$
Subtract the numerators and put the result on the LCM as follows:
$={\left[\frac{\omega {\omega }_{0}-\omega {\omega }_{0}}{\omega {\omega }_{0}}\right]}^{2}$
$={\left[\frac{0}{\omega {\omega }_{0}}\right]}^{2}$
Apply this exponent rule
$={\left[0\right]}^{2}=0$