Simplify (x²×y³×z⁴)²: Advanced Exponent Expression

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

$(a\cdot b)^n=a^n\cdot b^n$

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

Let's apply this law to our problem:

$(x^2\cdot y^3\cdot z^4)^2=(x^2)^2\cdot(y^3)^2\cdot(z^4)^2$

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

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

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

Let's apply this law to the expression we got:

$(x^2)^2\cdot(y^3)^2\cdot(z^4)^2=x^{2\cdot2}\cdot y^{3\cdot2}\cdot z^{4\cdot2}=x^4\cdot y^6\cdot z^8$

where in the second stage we performed the multiplication operation in the power exponents of the factors we obtained.

Therefore, the correct answer is answer D.

Note:

From the above formulation of the power law for parentheses, it might seem that it only refers to two factors in a product within parentheses, but in fact, it is valid for a power of a product of multiple factors in parentheses, as demonstrated in this problem and others,

It would be a good exercise to prove that if the above law is valid for a power of a product of two factors in parentheses (as it is formulated above), then it is also valid for a power of a product of multiple factors in parentheses (for example - three factors, etc.).