Solve ((x^(1/4) × 3² × 6³)^(1/4))^8: Complex Exponent Simplification

Proceed to solve this in two stages. In the first stage, we'll use the power rule for powers in parentheses:

$(z\cdot t)^n=z^n\cdot t^n$

which states that when a power is applied to terms in parentheses, it applies to each term inside the parentheses when they are opened,

Let's apply this rule to our problem:

$\big((x^{\frac{1}{4}}\cdot3^2\cdot6^3)^{\frac{1}{4}}\big)^8=((x^{\frac{1}{4}})^{\frac{1}{4}}\cdot(3^2)^{\frac{1}{4}}\cdot(6^3)^{\frac{1}{4}})^8$

When opening the parentheses, we applied the power to each term separately, however given that each of these terms is raised to a power, we did this carefully and used parentheses,

Next, we'll use the power rule for a power raised to a power:

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

Let's apply this rule to the expression that we obtained:

$(x^{\frac{1}{4}\cdot\frac{1}{4}}\cdot3^{2\cdot\frac{1}{4}}\cdot6^{3\cdot\frac{1}{4}})^8=(x^{\frac{1}{16}}\cdot3^{\frac{2}{4}}\cdot6^{\frac{3}{4}})^8=x^{\frac{1}{16}\cdot8}\cdot3^{\frac{2}{4}\cdot8}\cdot6^{\frac{3}{4}\cdot8}=x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}$

In the second stage we performed multiplication in the fractions of the power expressions of the terms that we obtained. Remember that multiplication in fractions is actually multiplication in the numerator. In the final stage we simplified the fractions in the power expressions of the multiplication terms that we obtained:

$x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}=x^{\frac{1}{2}}\cdot3^4\cdot6^6$

Therefore, the correct answer is answer B.