Simplify the Expression: a⁵ × aˣ × (abc)⁵ × (a¹)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 rule states that when an exponent is applied to parentheses containing a product of terms, the exponent is distributed to each factor inside the parentheses individually.

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

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

This rule 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.

Now, let’s return to the problem and handle the two terms inside the parentheses separately before continuing with the overall multiplication.

1. The first from left to right is:

$(abc)^5=a^5\cdot b^5\cdot c^5$

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

2. The second from left to right is:

$(a^1)^5=a^{1\cdot5}=a^5$

When we applied the power to a power (while opening the parentheses in practice) using the second law above.

Going back to the problem, we obtained the following:

$a^5\cdot a^x\cdot(abc)^5\cdot(a^1)^5=a^5\cdot a^x\cdot a^5\cdot b^5\cdot c^5\cdot a^5$

When we used 1 and 2 mentioned above.

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

Now let's arrange the expression by bases using the multiplication distributive law:

$a^5 a^x a^5 b^5 c^5 a^5 =a^5a^xa^5a^5b^5c^5$

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 the problem:

$a^5a^xa^5a^5b^5c^5=a^{5+x+5+5}b^5=a^{15+x}b^5c^5$

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 above power of a power law (the second law above), this law comes 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 the terms as a sum in the exponent,

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

$m+m+\cdots+m=m\cdot n$ and therefore we get that:

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