Simplify the Expression: a^7 × (abc)^3 × a^4 Using Exponent Rules

Solve the following problem:

$a^7\times(abc)^3\times a^4=$

Begin by carefully opening the parentheses, using the law of exponents that applies to parentheses containing a multiplication of terms:

$(z\cdot y)^n=z^n\cdot y^n$
This essentially states that when there is an exponent acting on parentheses containing multiplication between terms, the exponent will apply separately on each of the multiplication terms inside the parentheses.

Let's return to the problem and deal with the term containing parentheses in the overall multiplication separately-

$(abc)^3=a^3\cdot b^3\cdot c^3$
When we opened the parentheses using the law of exponents mentioned above, we applied the exponent to each of the multiplication terms inside of the parentheses.

Let's return to the problem as shown below:

$a^7\cdot(abc)^3\cdot a^4=a^7\cdot a^3\cdot b^3\cdot c^3\cdot a^4$
From here on we will no longer indicate the multiplication sign, instead of using the multiplication sign we will place terms next to each other to signify multiplication.

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

$a^7a^3b^3c^3a^4=a^7a^3a^4b^3c^3$
Proceed to 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 terms, for example for multiplication of three terms with the same base we obtain the following:

$y^m\cdot y^n\cdot y^k=y^{m+n}\cdot y^k=y^{m+n+k}$
Whilst we used the above law of exponents twice, we can also perform the same calculation for four terms in multiplication and so on...

Let's apply this law to the problem:

$a^7a^3a^4b^3c^3=a^{7+3+4}b^3c^3=a^{14}b^3c^3$
When we applied the above law of exponents, for multiplication between terms with identical bases only for terms with the same base.

We obtained the most simplified expression.

Therefore the correct answer is D.