Solve for X: x^5/16 = 2 Fifth-Power Equation

Solve the following problem:

$\frac{x^5}{16}=2$

Let's solve the given equation:

$\frac{x^5}{16}=2$

Begin by eliminating the denominator on the left side of the given equation, we can achieve this by multiplying both sides of the equation by the common denominator - which is the number 16:

$\frac{x^5}{16}=2 \hspace{8pt}\text{/}\cdot 16\\ x^5=2\cdot 16\\ x^5=32$

From here, proceed to perform on both sides the opposite operation to the fifth power that applies to the unknown in the equation, which is the fifth root operation, using the 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:

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

Let's continue solving the equation:
$x^5=32 \hspace{8pt}\text{/}\sqrt[5]{\hspace{6pt}}\\ \sqrt[5]{ x^5}=\sqrt[5]{ 32}\\ (x^5)^{\frac{1}{5}}=2\\ x^{5\cdot\frac{1}{5}}=2\\ \boxed{x=2}$

In the first stage, we applied the fifth root to both sides of the equation. Applying the definition of root as a power (a.) on the left side. In the next stage - we applied the law of exponents for a power of a power (b.) on the left side. Remember that raising a number to the power of 1 doesn't change the number.

Additionally, an odd-order power preserves the sign of the number it's applied to, taking an odd-order root requires consideration of only one case regarding the sign of the number being rooted (this is unlike taking an even-order root, which requires consideration of two possible cases - positive and negative),

Let's summarize the solution of the equation:

$\frac{x^5}{16}=2 \hspace{8pt}\text{/}\cdot 16\\ x^5=32 \hspace{8pt}\text{/}\sqrt[5]{\hspace{6pt}}\\ \boxed{x=2}$

Therefore, the correct answer is answer B.