Solve (x²×a³)^(1/4): Fourth Root of Multiple Variables

Let's solve this in two stages. In the first stage, we'll use the rule for a power of a product in parentheses:

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

This rule states states that, when raising a product in parentheses to a power, each factor in the product is raised to that power when expanding the parentheses.

Let's apply this rule to our problem:

$(x^2\cdot a^3)^{\frac{1}{4}}=(x^2)^{\frac{1}{4}}\cdot(a^3)^{\frac{1}{4}}$

For our problem, when opening the parentheses, we apply the power to each factor of the product separately; but since each of these factors is being raised to a power, we do this carefully and use parentheses.

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

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

Let's apply this rule to our expression:

$(x^2)^{\frac{1}{4}}\cdot(a^3)^{\frac{1}{4}}=x^{2\cdot\frac{1}{4}}\cdot a^{3\cdot\frac{1}{4}}=x^{\frac{2}{4}}\cdot a^{\frac{3}{4}}=x^{\frac{1}{2}}\cdot a^{\frac{3}{4}}$

In the second stage, we performed the multiplication in the exponents of the factors we obtained, while remembering that multiplying fractions means multiplying their numerators and then, in the final stage, we simplified the fraction in the power of the first factor in the resulting product.

Therefore, the correct answer is answer A.