How do you solve\[{{\left( -2{{a}^{2}} \right)}^{3}}\]?

Hint: In the given question we have been asked to find the value of\[{{\left( 2a \right)}^{3}}\]. In order to solve this question, first we need to use the laws of exponents and powers to solve the given question i.e. \[{{\left( ab \right)}^{m}}={{a}^{m}}\times {{b}^{^{m}}}\] and\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]. Using these properties of exponent we will solve the given question and simplify the result and we will get our required solution.

Formula Used: Using the laws of exponents which states that;
If ‘a’ and ‘b’ are two positive rational number and ‘m’ is the given rational exponent either positive exponent or negative exponent, then
\[{{\left( ab \right)}^{m}}={{a}^{m}}\times {{b}^{^{m}}}\]
\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]

Complete step-by-step solution:
We have given that,
\[\Rightarrow {{\left( -2{{a}^{2}} \right)}^{3}}\]
Using the laws of exponents which states that;
If ‘a’ and ‘b’ are two positive rational number and ‘m’ is the given rational exponent either positive exponent or negative exponent, then
\[{{\left( ab \right)}^{m}}={{a}^{m}}\times {{b}^{^{m}}}\]
Applying the law of exponent i.e. \[{{\left( ab \right)}^{m}}={{a}^{m}}\times {{b}^{^{m}}}\] in the above given question, we obtain
\[\Rightarrow {{\left( -2{{a}^{2}} \right)}^{3}}=-{{2}^{3}}\times {{\left( {{a}^{2}} \right)}^{3}}\]
Now,
\[\Rightarrow -{{2}^{3}}=-2\times -2\times -2=-8\]
Substituting \[{{2}^{3}}=8\] in the above equation, we get
\[\Rightarrow {{\left( -2{{a}^{2}} \right)}^{3}}=-8\times {{\left( {{a}^{2}} \right)}^{3}}\]
Now,
Using the laws of exponents which states that;
If ‘a’ and ‘b’ are two positive rational number and ‘m’ and ‘n’ are the given rational exponent either positive exponent or negative exponent, then
\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]
Applying this law of exponent in the above question, we get
\[\Rightarrow {{\left( -2{{a}^{2}} \right)}^{3}}=-8\times \left( {{a}^{2\times 3}} \right)\]
Simplifying the above equation, we get
\[\Rightarrow {{\left( -2{{a}^{2}} \right)}^{3}}=-8\times \left( {{a}^{6}} \right)\]
Therefore,
\[\Rightarrow {{\left( -2{{a}^{2}} \right)}^{3}}=-8{{a}^{6}}\]
It is the required possible answer.

Note: While solving these types of questions, students should remember all the laws of exponents as they will be able to solve the question easily. Students need to remember that also if a number is expressed in exponential form and has a negative exponent, then first we need to convert the negative exponent to a positive exponent by taking the reciprocal of the base. While applying any law of exponents we should very carefully write the terms in a respective form as it is given in the associated law to it.