Question

Simplify: - \[\dfrac{4+10\omega }{8-50{{\omega }^{2}}}\].

Answer 1

Hint: Take 2 common from both numerator and denominator of the given expression and cancel them. Now, apply the algebraic identity given as: \[\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)\] in the denominator and cancel the common term in the numerator with this common term of simplified denominator. Hence, write the simplified form of the given expression.

Complete step by step answer:
Here, we have been asked to simplify the expression: - \[\dfrac{4+10\omega }{8-50{{\omega }^{2}}}\].
Now, taking 2 common from both the numerator and the denominator of the given expression, we get,
\[\Rightarrow \dfrac{4+10\omega }{8-50{{\omega }^{2}}}=\dfrac{2\left( 2+5\omega \right)}{2\left( 4-25{{\omega }^{2}} \right)}\]
Cancelling the common factor 2, we get,
\[\Rightarrow \dfrac{4+10\omega }{8-50{{\omega }^{2}}}=\left( \dfrac{2+5\omega }{4-25{{\omega }^{2}}} \right)\] - (1)
Now, we can clearly see that the denominator of the above expression in the right - hand side is \[4-25{{\omega }^{2}}\] which can be written in the form: - \[{{a}^{2}}-{{b}^{2}}\] as given below,
\[\begin{align}
  & \Rightarrow 4-25{{\omega }^{2}}={{\left( 2 \right)}^{2}}-{{5}^{2}}{{\omega }^{2}} \\
 & \Rightarrow 4-25{{\omega }^{2}}={{2}^{2}}-{{\left( 5\omega \right)}^{2}} \\
\end{align}\]
So, substituting this value in equation (1), we get,
\[\Rightarrow \dfrac{4+10\omega }{8-50{{\omega }^{2}}}=\dfrac{2+5\omega }{{{2}^{2}}-{{\left( 5\omega \right)}^{2}}}\]
Now, we know that if we are provided with an algebraic expression of the form \[\left( {{a}^{2}}-{{b}^{2}} \right)\] then it can be simplified using the algebraic identity given as: - \[\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)\], so for the above obtained expression, we have,
\[\Rightarrow \dfrac{4+10\omega }{8-50{{\omega }^{2}}}=\dfrac{2+5\omega }{\left( 2+5\omega \right)\left( 2-5\omega \right)}\]
Now, cancelling the common term \[\left( 2+5\omega \right)\] from both numerator and denominator, we get,
\[\Rightarrow \dfrac{4+10\omega }{8-50{{\omega }^{2}}}=\dfrac{1}{\left( 2-5\omega \right)}\]
Hence, the simplified form of the given expression is: -
\[\Rightarrow \dfrac{4+10\omega }{8-50{{\omega }^{2}}}=\dfrac{1}{\left( 2-5\omega \right)}\], which is our required answer.

Note: One may note that whenever we are asked to simplify a given expression which is algebraic in nature then we use basically three formulas of algebraic identities according to the given situation. These identities are: - \[\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right),{{a}^{2}}+{{b}^{2}}-2ab={{\left( a-b \right)}^{2}}\], \[{{a}^{2}}+{{b}^{2}}+2ab={{\left( a+b \right)}^{2}}\]. In the above question we have used the first formula because the other two identities are not applicable for the given situation. Do not forget to cancel the common factor wherever possible.