Question

If \[a{\text{ }} = {\text{ }}\left( { - 2} \right),\]then find the value of \[{\left( {2a} \right)^5} \times {\left( {\dfrac{a}{2}} \right)^3} \div {\left( a \right)^4}\].

Answer 1

Hint: To solve this kind of question we are going to use the following procedure given below so that we can make our question solving process as simple as possible and that will be helpful to save our valuable time.
Apply the following exponent rules
\[
  {{x^n}.{\text{ }}{x^{m\;}} = {\text{ }}{x^{n + m}}} \\
  {\dfrac{{{x^n}}}{{{x^m}}}\; = {\text{ }}{x^{n - m}}} \\
  {{x^0} = {\text{ }}1} \\
  {{{\left( {xy} \right)}^n}\; = {\text{ }}{x^n}{y^n}}
\]

Complete step-by-step answer:
Given:If \[a{\text{ }} = {\text{ }}\left( { - 2} \right),\] is given.
When we are solving this type of question, we need to follow the steps provided in the hint part above.
We are given that
\[{\left( {2a} \right)^5} \times {\left( {\dfrac{a}{2}} \right)^3} \div {\left( a \right)^4}\].
Now we are going to use \[\dfrac{{{x^n}}}{{{x^m}}}\; = {\text{ }}{x^{n - m}}\] and \[{x^n}.{\text{ }}{x^{m\;}} = {\text{ }}{x^{n + m}}\].
Formula.
\[
\Rightarrow {\left( {2a} \right)^5} \times {\left( {\dfrac{a}{2}} \right)^3} \div {\left( a \right)^4} = {\left( {2a} \right)^5} \times \left( {\dfrac{{{a^{3 - 4}}}}{{{2^3}}}} \right) \\
   = {\left( {2a} \right)^5} \times \left( {{a^{ - 1}}{2^{ - 3}}} \right) \
 \]
Here we are going to apply the following formulas
\[{\left( {xy} \right)^n}\; = {\text{ }}{x^n}{y^n}\]
\[
\Rightarrow {\left( {2a} \right)^5} \times {\left( {\dfrac{a}{2}} \right)^3} \div {\left( a \right)^4} = {\left( {2a} \right)^5} \times \left( {{a^{ - 1}}{2^{ - 3}}} \right) \\
   = {2^{5 - 3}}{a^{5 - 1}} \\
   = {2^2}{a^4} \\
   = 4{\left( { - 2} \right)^4} \\
   = 4 \times 16 \\
   = 64 \
 \]
Here answer is \[64\]

Note: In this sort of examples, we need to face the various things and some of them are referred to here which will be really helpful to fathom the main concept of problem:
We need to use right formula to avoid unnecessary things which are sometime time consuming:
We need use the concept that
\[
  {{{\left( { - 1} \right)}^{odd{\text{ }}integer}} = {\text{ }} - 1} \\
  {{{\left( { - 1} \right)}^{even{\text{ }}integer}} = {\text{ }}1}
\]
At the end the subtitle \[a{\text{ }} = {\text{ }} - 2\]carefully.