To determine the value of [tex]\((-64)^{\frac{2}{3}}\)[/tex], we need to follow these steps:
1. Understand the properties of exponents and roots: The exponent [tex]\(\frac{2}{3}\)[/tex] means we should first take the cube root of [tex]\(-64\)[/tex] and then square the result. However, with a negative base and a rational exponent, we should expect a complex number.
2. Identify the cube root of [tex]\(-64\)[/tex]: The real cube root of [tex]\(-64\)[/tex] is [tex]\(-4\)[/tex] because [tex]\((-4)^3 = -64\)[/tex]. However, complex numbers will involve using complex cube roots as well.
3. Square the cube root: For a complex number [tex]\((-4)^{\frac{2}{3}}\)[/tex], squaring [tex]\(-4\)[/tex] involves complex analysis where:
- We can rewrite [tex]\(-64\)[/tex] in polar form: [tex]\(64e^{i\pi}\)[/tex].
- Using De Moivre's theorem and properties of exponents, [tex]\((-64)^{2/3} = (64e^{i\pi})^{2/3} = 64^{2/3} \cdot (e^{i\pi})^{2/3}\)[/tex].
- [tex]\(64^{2/3} = 16\)[/tex].
- [tex]\((e^{i\pi})^{2/3} = e^{2i\pi/3}\)[/tex], which corresponds to [tex]\(\pi/3\)[/tex] away from the initial position on the Argand plane, giving us the primary branch [tex]\(5\pi/3\)[/tex] and [tex]\(-\pi/3\)[/tex].
- Therefore, [tex]\(16 \cdot e^{2i\pi/3}\)[/tex] falls within:
Thus, the value of the expression [tex]\((-64)^{\frac{2}{3}}\)[/tex] in complex form turns out to be [tex]\((-7.999999999999996 + 13.856406460551018j)\)[/tex].
This does not directly align with any exact value from the given multiple-choice answers, so based on this complex form, there is no straightforward match to the provided options [tex]\(-16\)[/tex], [tex]\(4\)[/tex], [tex]\(16\)[/tex], or [tex]\(-4\)[/tex]. Perhaps the question might include a misunderstanding or they need to consider complex number solutions.
