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

Solve the following problem:

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

Let's solve the given equation:

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

We will perform the inverse operation of the cube power (applied to the unknown) on both sides of the equation. This is the cube root operation. We will need to apply 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 to solve 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. Apply the definition of root as a power (a.) on the left side as well as the law of exponents for power of parentheses (c.) on the right side. Note that raising 1 to any power always yields 1. In the next step, we applied the law of exponents for power of power (b.) Remember that raising a number to the power of 1 doesn't change the number,

Furthermore, given that 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.