How do you multiply \[{\left( {2{a^{ - 2}}} \right)^3}{\left( {{a^4}{a^{ - 5}}} \right)^6}\] ?

Hint: Here we have a simple algebraic expression. We can simplify this using the laws of indices. Without knowing the laws of indices it is impossible to simplify any mathematical equation or expression. There are several basic law of indices, namely law of multiplication, law of division, law of bracket, the power rule, negative power and fractional power. We use these to solve the given problem.

Complete step-by-step answer:
Given, \[{\left( {2{a^{ - 2}}} \right)^3}{\left( {{a^4}{a^{ - 5}}} \right)^6}\] .
Using the multiplication law that is \[ \Rightarrow {x^m} \times {x^n} = {x^{m + n}}\] , the terms in the second parentheses becomes
 \[ \Rightarrow {\left( {2{a^{ - 2}}} \right)^3}{\left( {{a^{4 - 5}}} \right)^6}\]
 \[ \Rightarrow {\left( {2{a^{ - 2}}} \right)^3}{\left( {{a^{ - 1}}} \right)^6}\] .
 \[ \Rightarrow {\left( 2 \right)^3}{\left( {{a^{ - 2}}} \right)^3}{\left( {{a^{ - 1}}} \right)^6}\]
Now applying bracket law that is \[ \Rightarrow {\left( {{x^m}} \right)^n} = {x^{m \times n}}\] , we have,
 \[ \Rightarrow 8\left( {{a^{ - 2 \times 3}}} \right)\left( {{a^{ - 1 \times 6}}} \right)\]
 \[ \Rightarrow 8\left( {{a^{ - 6}}} \right)\left( {{a^{ - 6}}} \right)\]
Again applying the multiplication law,
 \[ \Rightarrow 8\left( {{a^{ - 6 - 6}}} \right)\]
 \[ \Rightarrow 8\left( {{a^{ - 12}}} \right)\] .
Hence the multiplication of \[{\left( {2{a^{ - 2}}} \right)^3}{\left( {{a^4}{a^{ - 5}}} \right)^6}\] is \[ \Rightarrow 8\left( {{a^{ - 12}}} \right)\]

Note: We have several laws of indices.
 \[ \bullet \] The first law: multiplication: if the two terms have the same base and are to be multiplied together their indices are added. That is \[ \Rightarrow {x^m} \times {x^n} = {x^{m + n}}\]
 \[ \bullet \] The second law: division: If the two terms have the same base and are to be divided their indices are subtracted. That is \[ \Rightarrow \dfrac{{{x^m}}}{{{x^n}}} = {x^{m - n}}\]
 \[ \bullet \] The third law: brackets: If a term with a power is itself raised to a power then the powers are multiplied together. That is \[ \Rightarrow {\left( {{x^m}} \right)^n} = {x^{m \times n}}\]
 \[ \bullet \] As we have the second law of indices which helps to explain why anything to the power of zero is equal to one. \[ \Rightarrow {x^0} = 1\]
 \[ \bullet \] Negative power \[ \Rightarrow {x^{ - n}} = \dfrac{1}{{{x^m}}}\]
 \[ \bullet \] The fractional power \[ \Rightarrow {x^{\dfrac{m}{n}}} = \left( {\sqrt[n] {m}} \right)\] . We use them depending on the given expression.