Question

How do you evaluate ${( - 1)^{\dfrac{2}{3}}}$?

Answer 1

Hint:
Whenever we are given a problem on simplification of powers of a number then we can use exponent or radical rule formula to simplify and get the correct answer. Exponent or radical rule formula is given by: ${x^{\dfrac{m}{n}}} = \sqrt[n]{{{x^m}}}$. By substituting the corresponding values in the formula we can arrive at the solution.

Complete Step by Step Solution:
The problem of powers or exponent can be solved by using exponent or radical rule formula which is given by:
${x^{\dfrac{m}{n}}} = \sqrt[n]{{{x^m}}}$
Where $x$ is a number
$\dfrac{m}{n}$ value of exponent
In the given question they have asked us to solve for ${( - 1)^{\dfrac{2}{3}}}$.
Here, from the given problem we can notice that $x$ is $ - 1$ , $m$ is given as $2$ and the value of $n$ is $3$.
Now, by substituting the values of $x$, $m$ and $n$ in the exponent or radical rule formula, we get
${( - 1)^{\dfrac{2}{3}}} = \sqrt[3]{{{{( - 1)}^2}}}$
We know that whenever we multiply negative with negative it gives positive that is $ - \times - = + $ . When we multiply $1$ with any other number we get the answer as the number itself. In the same way if we multiply one with one then the result will be one itself. So we can reduce the above equation using these concepts.
Therefore, we get
$ \Rightarrow {( - 1)^{\dfrac{2}{3}}} = \sqrt[3]{1}$
As when we multiply one with one we get one, in the same way if we do cube root of one the answer will be one only.
$ \Rightarrow {( - 1)^{\dfrac{2}{3}}} = 1$

Therefore the answer is $1$.

Note:
If we have a negative sign with $1$ it becomes positive only when we have an even number of power raised to that number, if it is an odd number then the sign remains the same. Only in case of number one we have like if multiply with any number the answer is the number itself but it is not the case with any other numbers.