Solve Complex Expression: a⁴ × a³ × (abc)⁸ × ((a³)⁴)⁵

Let's begin by carefully opening the parentheses, using two laws of exponents:

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

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

This law states that when an exponent is applied to parentheses containing a product of terms, the exponent is distributed to each factor inside the parentheses.

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

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

This law states that when an exponent is applied to a term that is already raised to a power (whether written with parentheses for clarity or not), the exponents are multiplied together.

Let’s now return to the problem and first simplify the two parenthetical terms separately before proceeding with the overall multiplication.

1. The first from left to right is:

$(abc)^8=a^8\cdot b^8\cdot c^8$

When we expanded the parentheses using the first law, we applied the exponent to each factor in the product inside the parentheses.

2. The second from left to right is:

$\big((a^3)^4\big)^5=(a^3)^{4\cdot5}=a^{3\cdot4\cdot5}=a^{60}$

When we expanded the parentheses step by step, moving from the outside inward, we first applied the power of a powerrule to the outer parentheses. Then, in the next stage, we applied the same rule again to the remaining expression inside.

In general, since a power of a power is interpreted as multiplying the exponents (according to the second law above), when we encounter repeated exponents—a power raised to another power—we simply multiply all the exponents together in the final expression.

Returning to the problem, we obtained the following:

$a^4\cdot a^3\cdot(abc)^8\cdot\big((a^3)^4\big)^5= a^4\cdot a^3\cdot a^8\cdot b^8\cdot c^8\cdot a^{60}$

Where we used 1 and 2 noted above.

From here on we will no longer indicate the multiplication sign, instead using the conventional writing form where placing terms next to each other means multiplication.

Now let's arrange the expression by bases while using the multiplication property of exponents:

$a^4 a^3 a^8 b^8 c^8 a^{60} =a^4 a^3 a^8 a^{60} b^8 c^8$

Let's continue and simplify the expression by using the law of exponents for multiplication between terms with identical bases:

$y^m\cdot y^n=y^{m+n}$

Note that this law is valid for any number of terms in multiplication and not just for two, for example for multiplication of three terms with identical bases we get:

$y^m\cdot y^n\cdot y^k=y^{m+n}\cdot y^k=y^{m+n+k}$

When we used the above exponent law twice, we can also perform the same calculation for four terms in multiplication five etc..

Let's apply this law to our problem:

$a^4 a^3 a^8 a^{60} b^8 c^8 =a^{4+3+8+60}b^8 c^8=a^{75}b^8c^8$

When we used the above exponent law, for multiplication between terms with identical bases only for terms with the same base.

We got the most simplified expression, so we're done.

Therefore the correct answer is A.

Important note:

It's worth understanding the reason for the power of power law mentioned above (the second law), this law stems directly from the definition of exponents:

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

When 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 above law of exponents for multiplication between terms with identical bases and interpreted the multiplication between terms as a sum in the exponent notation,

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

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

And therefore we get that:

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