Solve ((9xyz)^3)^4 + (a^y)^x: Complex Compound Exponents

We'll use the power rule for a power:

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

We'll apply this rule to the expression in the problem in two stages:

$((9xyz)^3)^4+(a^y)^x= (9xyz)^{3\cdot4}+(a^y)^x=(9xyz)^{12}+a^{yx}$

In the first stage, we apply the above rule initially to the first term in the expression and then proceed to deal with the outer parentheses. We then simplify the expression in the exponent whilst simultaneously applying the power rule to the second term in the sum in the problem's expression.

We'll continue by recalling the rule for powers that applies to parentheses containing the multiplication of terms:

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

We'll apply this rule to the expression that we obtained in the last stage:

$(9xyz)^{12}+(a^y)^x =9^{12} x^{12} y^{12}z^{12}+a^{yx}$

We apply the aforementioned power rule to the first term in the sum in the expression that we obtained in the last stage, and apply the power on the parentheses to each of the multiplication terms inside the parentheses.

Let's summarize the solution steps so far:

$((9xyz)^3)^4+(a^y)^x=(9xyz)^{12}+a^{yx} =9^{12} x^{12} y^{12} z^{12}+a^{yx}$

Therefore the correct answer is answer D.