Simplify ((5a)²)³ + (xyz)^(1/4): Complex Exponent Expression

First, we'll carefully expand the parentheses, using two laws of exponents:

The first law is the exponent law that applies to parentheses containing multiplication of terms:

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

This law essentially states that when there is an exponent acting on parentheses containing multiplication between terms, when opening the parentheses the exponent will apply separately to each of the multiplication terms inside the parentheses.

The second law we'll use is the power of a power law:

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

This law states that when applying an exponent to a term that is already raised to a power we can interpret this as multiplication between the exponents within the exponent notation.

Let's return to the problem and first deal with the two parenthetical terms in the overall sum separately:

1. The second from left to right is:

$(xyz)^{\frac{1}{4}}=x^{\frac{1}{4}}y^{\frac{1}{4}}z^{\frac{1}{4}}$

We open the parentheses using the first law mentioned above, so that when opening the parentheses we apply the exponent to each of the multiplication terms inside the parentheses.

2. The first from left to right is:

$\big((5a)^2\big)^3=(5^2a^2)^3=(5^2)^3(a^2)^3=5^{2\cdot3}a^{2\cdot3}=5^6a^6$

We used the first law above twice, first for the inner parentheses and then for the remaining parentheses, but we did this carefully since the terms in the multiplication within the parentheses are raised to powers and therefore we performed this using additional parentheses; and then we applied the power to the power (while effectively opening the parentheses) using the second law above.

Going back to the problem, we have:

$\big((5a)^2\big)^3+(xyz)^{\frac{1}{4}}=5^6a^6+ x^{\frac{1}{4}}y^{\frac{1}{4}}z^{\frac{1}{4}}$

We use 1 and 2 that we noted above.

We now have the most simplified expression.

Therefore the correct answer is C.

Important note:

It's worth noting the reasoning for the power of a power law mentioned above (the second law). This law comes directly from the definition of exponents:

$(c^m)^n=c^m\cdot c^m\cdot\ldots\cdot c^m=c^{m+m+m+\cdots+m}=c^{m\cdot n}$

In the first stage we applied the definition of exponents to the term in parentheses and multiplied it by itself n times, then we applied the law of exponents for multiplication between terms with identical bases mentioned above and interpreted the multiplication between the terms as a sum in the exponent.

Then we used the simple multiplication definition that says if we connect a number to itself n times we can simply write this as multiplication, meaning:

$m+m+\cdots+m=m\cdot n$

Therefore:

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