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, for good order, we applied the above power rule first to the first term in the expression and dealt with the outer parentheses, then we simplified the expression in the exponent while simultaneously applying the mentioned power rule to the second term in the sum in the problem's expression,

We'll continue and recall the power rule for powers that applies to parentheses containing multiplication of terms:

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

We'll apply this power rule to the expression we got in the last stage:

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

When we applied the mentioned power rule to the first term in the sum in the expression we got in the last stage, and applied the power on the parentheses to each of the multiplication terms inside the parentheses,

Let's summarize the solution steps so far, we got that:

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

Therefore the correct answer is answer D.