Evaluate: ${\left( {{1^3} + {2^3} + {3^3}} \right)^{\dfrac{1}{2}}}$

Hint: We know that the above-given expression is in exponential form. We know from the definition that an exponent refers to the number of times a number is multiplied by itself. There is base and exponent or power in these types of equations. Outside the bracket, here, in the given question $\left( {{1^3} + {2^3} + {3^3}} \right)$ is the base and the number $\dfrac{1}{2}$ is the exponential power. While inside of the bracket we can say that $(1 + 2 + 3)$ is the base and they have the same common power i.e., $3$ . We will now solve the values and simplify them.

Complete answer:
Here we have the expression ${\left( {{1^3} + {2^3} + {3^3}} \right)^{\dfrac{1}{2}}}$ .
We will solve the inside of the bracket first.
 And we know that
 ${1^3} = 1,{2^3} = 8$
And ${3^3}$ gives the value
$3 \times 3 \times 3 = 27$ .
Therefore by substituting the values, we have:
${\left( {1 + 8 + 27} \right)^{\dfrac{1}{2}}}$
It gives the value
${\left( {36} \right)^{\dfrac{1}{2}}}$
We can write $36$ as ${6^2}$, so the above expression can also be written as :
${\left( {{6^2}} \right)^{\dfrac{1}{2}}}$ .
As we know that another rule of exponent says that
${({a^m})^p} = {a^{m \times p}}$,
So by applying the formula of exponential power equations in the equation, we can write:
$\left( {{6^{2 \times \dfrac{1}{2}}}} \right)$.
It gives us an answer of $6$.
Hence the required answer is $6$

Note: We know that exponential equations are those equations in which variables occur as an exponent. We should solve this kind of problem by using an exponential formula to simplify the problem. We should keep in mind while solving questions that if there is a negative value in the power or exponent then it will reverse the number i.e., ${m^{ - x}}$ will always be equal to. When we express a number in exponential form then we can say that its power has been raised by the exponent.