Question

Simplify \[\dfrac{{36 - \dfrac{1}{{{x^2}}}}}{{\dfrac{1}{{6{x^2}}} - 6}}\]

Answer 1

Hint: To proceed, first simplify numerator and denominator separately by cross multiplication method. Then combine both numerator and denominator again and find the expression that comes out. Then, cancel the common terms to find a simplified value.

Complete Step by Step Solution:
Now, let us simplify the numerator and denominator in the given question:-
\[\Rightarrow\dfrac{{36 - \dfrac{1}{{{x^2}}}}}{{\dfrac{1}{{6{x^2}}} - 6}}\]\[\]
Simplifying the numerator, we get:
$\Rightarrow36 - \dfrac{1}{{{x^2}}} = \dfrac{{36{x^2} - 1}}{{{x^2}}}$
Simplifying the denominator, we get:
\[\Rightarrow\dfrac{1}{{6{x^2} - 6}} = \dfrac{{1 - 36{x^2}}}{{6{x^2}}}\]
Now,
Again combining the numerator and denominator, we get
$\Rightarrow\dfrac{{\dfrac{{\left( {36{x^2} - 1} \right)}}{{{x^2}}}}}{{\dfrac{{\left( {1 - 36{x^2}} \right)}}{{6{x^2}}}}} = \dfrac{{36{x^2} - 1}}{{{x^2}}} \times \dfrac{{\left( {6{x^2}} \right)}}{{\left( {1 - 36{x^2}} \right)}}...(i)$ (By cross multiplication method)
Now, let us Multiply the equation $(i)$by ( -1 ) and divide by ( -1 )
$ \Rightarrow \dfrac{{\left( { - 1} \right)}}{{\left( { - 1} \right)}}\dfrac{{\left( {36{x^2} - 1} \right)}}{{{x^2}}} \times \dfrac{{\left( {6{x^2}} \right)}}{{\left( {1 - 36{x^2}} \right)}} = \left( { - 1} \right) \times \dfrac{{\left( {36{x^2} - 1} \right)}}{{{x^2}}} \times \dfrac{{\left( {6{x^2}} \right)}}{{\left( { - 1} \right)\left( {1 - 36{x^2}} \right)}}$
$ = \left( { - 1} \right)\dfrac{{\left( {36{x^2} - 1} \right)}}{{{x^2}}} \times \dfrac{{\left( {6{x^2}} \right)}}{{\left( {36{x^2} - 1} \right)}} = - 6$
Therefore, $\dfrac{{36 - \dfrac{1}{{{x^2}}}}}{{\dfrac{1}{{6{x^2}}} - 36}} = - 6$, which is the required answer of the above given question.
Hence, On simplification of the above expression, we get the answer as -6(which is a digit and not a variable).

Additional information:
Sometimes, it is not at all necessary to get a simplified answer in terms of a number(which can be positive or negative both). You can also get answers in terms of variables such as x.A answer such as infinity, undefined or zero is also possible. You just need to solve correctly to get to the right answer.
Also, you sometimes need to use some basic mathematical identities to get to the simplified version.
Some of these identities are:-
\[\Rightarrow{a^2} - {b^2} = (a - b)(a + b)\]
\[\Rightarrow{(a + b)^2} = {a^2} + {b^2} + 2ab\]
\[\Rightarrow{(a - b)^2} = {a^2} + {b^2} - 2ab\]
 By using the above given identities, you can solve many questions.

Note: While solving such a fraction, we must be careful in cross multiplication steps. To simplify the cross-multiplication we can derive it as $\dfrac{{\dfrac{A}{B}}}{{\dfrac{C}{D}}} = \dfrac{A}{B} \times \dfrac{D}{C} = \dfrac{{AD}}{{BC}}$
For example:-
\[\dfrac{\left(\dfrac{2}{5}\right)}{\left(\dfrac{7}{9}\right)} = \dfrac{{2.9}}{{5.7}}\]\[ = \dfrac{{18}}{
  35 \\
    \\
 }\](Here, a=2, b=5,c=7, d=9 )