Which expression is equivalent to ${\left[ {\sqrt {{a^2}{b^{\dfrac{1}{2}}}} } \right]^{ - 1}}$?
A. ${a^{ - 2}}{b^{\dfrac{{ - 1}}{2}}}$
B. $ - a{b^{\dfrac{1}{4}}}$
C. $a{b^2}$
D. $\dfrac{1}{{a{b^{\dfrac{1}{4}}}}}$

Hint: Here, we have to determine which expression is equivalent to ${\left[ {\sqrt {{a^2}{b^{\dfrac{1}{2}}}} } \right]^{ - 1}}$. We have given an exponential expression. Exponents can be defined as the power or indices of a number. We will use law of exponents and power to solve the question which is ${\left( {{r^m}} \right)^n} = {r^{mn}}$ here, $r$ is the base and $m$ and $n$ are the exponents.

Complete step by step answer:
Exponents can be defined as the power or indices of a number and they tell us how many times the number is repeated or they can be defined as the numbers that are raised to the power of another number.

We will use law of exponents and power to solve the given exponential expression and the law states that when a number is raised by another number, the powers are multiplied with each other and it can be written as ${\left( {{r^m}} \right)^n} = {r^{m \times n}}$ where, $r$ is the base and $m$ and $n$ are the exponents. We have the expression ${\left[ {\sqrt {{a^2}{b^{\dfrac{1}{2}}}} } \right]^{ - 1}}$. We can write the above expression as ${\left( {{{\left( {{a^2}{b^{\dfrac{1}{2}}}} \right)}^{\dfrac{1}{2}}}} \right)^{ - 1}}$

Now, using the law of exponents i.e., ${\left( {{r^m}} \right)^n} = {r^{m \times n}}$. We get,
$ \Rightarrow {\left( {{a^{2 \times \dfrac{1}{2}}}{b^{\dfrac{1}{2} \times \dfrac{1}{2}}}} \right)^{ - 1}}$
Solving the powers of the expression. We get,
$ \Rightarrow {\left( {a{b^{\dfrac{1}{4}}}} \right)^{ - 1}}$
The above expression can be written as
$ \Rightarrow {a^{ - 1}}{b^{\dfrac{{ - 1}}{4}}}$
Further the above equation can be reduced by solving the negative powers.
$ \Rightarrow \dfrac{1}{{a{b^{\dfrac{1}{4}}}}}$
Therefore, the expression ${\left[ {\sqrt {{a^2}{b^{\dfrac{1}{2}}}} } \right]^{ - 1}}$ is equivalent to $\dfrac{1}{{a{b^{\dfrac{1}{4}}}}}$.

Hence, option D is the correct answer.

Note: In these types of questions, solve each power of base very carefully, otherwise the answer may get wrong. The power of a number is known as the exponent and is usually expressed as a raised number or raised symbol and in other words it can be defined as how many times we can multiply the number. For example- ${2^5}$, here $5$ is the exponent, and it can also be written as $2 \times 2 \times 2 \times 2 \times 2$.