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 power rule for a power of a product in parentheses:

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

which states that when raising a product in parentheses to a power, each factor in the product is raised to that power when opening the parentheses,

Let's apply this rule to our problem:

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

where when opening the parentheses, we applied the power to each factor of the product separately, but since each of these factors is being raised to a power, we did this carefully and used 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 the expression we got:

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

where 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.