Solve the Quartic Equation: x^4/16 = 1

Solve the following problem:

$\frac{x^4}{16}=1$

Let's solve the given equation:

$\frac{x^4}{16}=1$

Perform the inverse operation of the fourth power (that applies to the unknown) on both sides of the equation. This is the fourth 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 a power:

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

c. Law of exponents for power applied to a product in parentheses:

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

Proceed to solve the equation:

$\frac{x^4}{16}=1 \hspace{8pt}\text{/}\sqrt[4]{\hspace{6pt}}\\ \frac{\sqrt[4]{x^4}}{\sqrt[4]{16}}=\pm\sqrt[4]{ 1}\\ \frac{(x^4)^{\frac{1}{4}}}{2}=\pm1\\ \frac{x^{4\cdot\frac{1}{4}}}{2}=\pm1\\ \frac{x}{2}=\pm1\hspace{8pt}\text{/}\cdot 2\\ \boxed{x=2,-2}$

In the first stage, we applied the fourth root to both sides of the equation. We then applied the definition of root as a power (a.) on the left side and the law of exponents for power applied to a product in parentheses (c.). Raising 1 to any power always yields 1. In the next stage, we applied the law of exponents for power of a power (b.) to the fraction's numerator on the left side, and remembered that raising a number to the power of 1 doesn't change the number. Finally, we eliminated the fraction on the left side by multiplying both sides of the equation by the common denominator - which is the number 2.

Furthermore, due to the fact that an even power doesn't maintain the sign of the number it's applied to (it will always give a positive result), taking an even root of both sides of the equation requires considering two possible cases - positive and negative (this is unlike taking a root of an odd order, which requires considering only one case that matches the sign of the number the root is applied to),

Therefore, the correct answer is answer c.