To find the value of the expression [tex]\((-8)^{4 / 3}\)[/tex], we need to evaluate the exponentiation operation for a negative base raised to a fractional exponent.
1. Understanding the Expression:
The given expression is [tex]\((-8)^{4 / 3}\)[/tex]. The fractional exponent [tex]\(\frac{4}{3}\)[/tex] can be interpreted as a combination of taking the cube root and raising to the power of 4.
- First, we take the cube root of -8.
- Then, we raise the result to the power of 4.
2. Step-by-Step Calculation:
Let's break it down into these steps:
- Cube Root: The cube root of [tex]\(-8\)[/tex] is [tex]\(-2\)[/tex]. This is because [tex]\((-2) \times (-2) \times (-2) = -8\)[/tex].
- Raise to Power 4: Next, we raise [tex]\(-2\)[/tex] to the power of 4.
So, [tex]\((-2)^4\)[/tex] results in:
[tex]\[
(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16
\][/tex]
However, since we are dealing with complex numbers and the cube root of a negative number can also include complex roots, the actual evaluation involves considering all cube roots and raising them to the 4th power.
The value of [tex]\((-8)^{4/3}\)[/tex] in its complete form, considering complex numbers, is
[tex]\[
(-8.000000000000005-13.856406460551014j)
\][/tex]
This is a complex number result. Therefore, the expression does not equate to a simple real number presented in the options A through D.
Thus, given the complex result, none of the provided options (A. [tex]$-\frac{32}{3}$[/tex], B. [tex]$\frac{32}{3}$[/tex], C. -16, D. 16) are correct. The answer, considering the expressions properly with complex results, is indeed
[tex]\[
(-8.000000000000005-13.856406460551014j).
\][/tex]
Therefore, the correct interpretation reveals that our current choices are not sufficient to capture the true solution, which is inherently complex.
