Answer :
To determine the value of the expression \((-8)^{2/3}\), we need to understand the operations involved.
First, let's break down the expression:
1. Base and Exponent: The expression is \((-8)^{2/3}\). Here, the base is \(-8\) and the exponent is \(\frac{2}{3}\).
2. Interpretation of the Exponent: The exponent \(\frac{2}{3}\) can be interpreted as taking the cube root of \(-8\) and then squaring the result.
- Cube Root Calculation: The cube root of a number \(a\), written as \(\sqrt[3]{a}\), is a value \(x\) such that \(x^3 = a\). For our base \(-8\), the cube root can be found as follows:
[tex]\[ \sqrt[3]{-8} = -2 \][/tex]
since \((-2)^3 = -8\).
- Squaring: After finding the cube root, we need to square the result. Squaring \(-2\) gives:
[tex]\[ (-2)^2 = 4 \][/tex]
In typical arithmetic operations involving real numbers, the expectation is that this process would conclude that \((-8)^{2/3} = 4\).
However, considering the result provided:
[tex]\[ (-8)^{2/3} = (-1.999999999999999 + 3.4641016151377544j) \][/tex]
This result indicates that the value is a complex number, with a real part approximately \(-2\), and an imaginary part approximately \(3.464\). This complex result stems from the mathematical properties that when dealing with negative bases raised to fractional exponents, the calculations can introduce complex components, derived from De Moivre's theorem and Euler's formula.
Given the computed result, none of the provided answer choices corresponds exactly to the complex value obtained.
Thus, we should acknowledge that the resulting value \((-1.999999999999999 + 3.4641016151377544j)\) doesn't match any of the given options but can be understood as an approximate outcome due to the nature of complex numbers in this context.
The correct value of the given expression is not incorporated in the provided options A, B, C, or D.
First, let's break down the expression:
1. Base and Exponent: The expression is \((-8)^{2/3}\). Here, the base is \(-8\) and the exponent is \(\frac{2}{3}\).
2. Interpretation of the Exponent: The exponent \(\frac{2}{3}\) can be interpreted as taking the cube root of \(-8\) and then squaring the result.
- Cube Root Calculation: The cube root of a number \(a\), written as \(\sqrt[3]{a}\), is a value \(x\) such that \(x^3 = a\). For our base \(-8\), the cube root can be found as follows:
[tex]\[ \sqrt[3]{-8} = -2 \][/tex]
since \((-2)^3 = -8\).
- Squaring: After finding the cube root, we need to square the result. Squaring \(-2\) gives:
[tex]\[ (-2)^2 = 4 \][/tex]
In typical arithmetic operations involving real numbers, the expectation is that this process would conclude that \((-8)^{2/3} = 4\).
However, considering the result provided:
[tex]\[ (-8)^{2/3} = (-1.999999999999999 + 3.4641016151377544j) \][/tex]
This result indicates that the value is a complex number, with a real part approximately \(-2\), and an imaginary part approximately \(3.464\). This complex result stems from the mathematical properties that when dealing with negative bases raised to fractional exponents, the calculations can introduce complex components, derived from De Moivre's theorem and Euler's formula.
Given the computed result, none of the provided answer choices corresponds exactly to the complex value obtained.
Thus, we should acknowledge that the resulting value \((-1.999999999999999 + 3.4641016151377544j)\) doesn't match any of the given options but can be understood as an approximate outcome due to the nature of complex numbers in this context.
The correct value of the given expression is not incorporated in the provided options A, B, C, or D.
