Multiplication of Powers: Using the commutative law

First, we'll use the distributive property of multiplication and arrange the algebraic expression according to like bases:

$a^4a^5b^5$

Next, we'll use the laws of exponents to multiply terms with like bases:

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

Therefore, we can combine all terms with the same base under one base:

$a^{4+5}b^5=a^9b^5$

Note that we could only combine terms with identical bases using this law,

From here we can observe that the expression cannot be simplified further, and therefore this is the correct answer, which is answer B (since the distributive property of multiplication is satisfied).

Important Note:

Note that for multiplication between numerical terms, we can denote the multiplication operation using a dot ($\cdot$), known as dot-product, or using the "times" symbol ($\times$) known as cross-product. For numerical terms, these operations are identical. We can also indicate multiplication by placing the terms next to each other without explicitly writing the operation between them. In such cases, there is a universal understanding that this represents multiplication between the terms. Usually, the multiplication is not explicitly noted (meaning the last option we mentioned here), and if it is noted, dot notation is typically used. In this problem, both in the question and answer, they chose to use cross notation, but the meaning is always the same since we are dealing with numerical terms.

Begin by applying the distributive property of multiplication in order to arrange the algebraic expression according to like terms:

$x^3x^4x^{3+y}y^4z^6$

$$Next, we'll use the law of exponents to multiply terms with the same base:

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

Note that this law applies to any number of terms being multiplied, not just two. For example, when multiplying three terms with the same base, we obtain the following:

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

Therefore, we can combine all terms with the same base under one base:

$x^{3+4+3+y}y^4z^6=x^{10+y}y^4z^6$

In the second step we simply added the exponents together.

Note that we could only combine terms with the same base using this law,

From here we can see that the expression cannot be simplified further, and therefore this is the correct and final answer which is answer D.

Begin by applying the distributive property of multiplication and proceed to arrange the algebraic expression according to like bases:

$a^ya^xa^{6\text{ }}b^97^y$

Next, we'll use the power rule to multiply terms with the same base:

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

Note that this rule applies to any number of terms in multiplication, not just two. For example, when multiplying three terms with the same base, we obtain the following:

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

Therefore, we can combine all terms with the same base under one base:

$a^{y+x+6}b^97^y$

Note that we could only combine terms with identical bases using this rule,

From here we can see that the expression cannot be simplified further, and therefore this is the correct answer, which is answer C (since the distributive property of multiplication holds).