Solve the Cubic Equation: Finding x When x³ = 1/8

Let's solve the given equation:

$x^3=\frac{1}{8}$

Simply, we will perform on both sides the inverse operation of the cube power applied to the unknown in the equation, which is the cube root operation. We'll use several laws of exponents:

a. Definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

and two laws of exponents:

b. Law of exponents for power of power:

$(a^m)^n=a^{m\cdot n}$

c. Law of exponents for power of parentheses:

$\big(\frac{a}{b}\big)^n=\frac{a^n}{b^n}$

Let's proceed with solving the equation:
$x^3=\frac{1}{8} \hspace{8pt}\text{/}\sqrt[3]{\hspace{6pt}}\\ \sqrt[3]{x^3}=\sqrt[3]{ \frac{1}{8}}\\ \sqrt[3]{x^3}= \frac{\sqrt[3]{1}}{\sqrt[3]{ 8}}\\ (x^3)^{\frac{1}{3}}=\frac{1}{\sqrt[3]{ 8}}\\ x^{3\cdot\frac{1}{3}}=\frac{1}{2}\\ \boxed{x=\frac{1}{2}}$

In the first step, we applied the cube root to both sides of the equation. Then we recalled the definition of root as a power (a.) on the left side and the law of exponents for power of parentheses (c.) on the right side. Additionally, we remembered that raising 1 to any power always yields 1. In the next step, we applied the law of exponents for power of power (b.), and remembered that raising a number to the power of 1 doesn't change the number,

Furthermore, we remembered that since an odd power preserves the sign of the number it's applied to, taking an odd root requires considering only one possible case which matches the sign of the number being rooted (this is unlike taking an even root, which requires considering two possible cases - positive and negative),

Therefore, the correct answer is answer d.