Simplify the Expression: ax × 3ay × a2 × 2a Using Exponent Rules

Note that there is multiplication operation between all terms in the expression, hence we'll first apply the distributive property of multiplication in order to handle the coefficients of terms raised to powers, and the terms themselves separately. For greater clarity, let's break this down into steps:

$a^x\cdot3a^y\cdot a^2\cdot2a=3\cdot2\cdot a^x\cdot a^y\cdot a^2\cdot a=6\cdot a^x\cdot a^y\cdot a^2\cdot a$

Due to the multiplication operation between all terms we could do this, it should be noted that we can (and it's preferable to) skip the middle step, meaning:

Write directly:$a^x\cdot3a^y\cdot a^2\cdot2a=6\cdot a^x\cdot a^y\cdot a^2\cdot a$

From here on we won't write the multiplication sign anymore instead we simply place the terms next to each other\ place the term next to its coefficient to indicate multiplication between them,

Proceed to apply the law of exponents for multiplication of terms with identical bases:

$c^m\cdot c^n=c^{m+n}$Note also that this law applies to any number of terms being multiplied and not just two, for example for three terms with identical bases we obtain:

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

Whilst we used law of exponents twice, we can also perform the same calculation for four terms or 5 and so on..,

Let's return to the problem, and apply the above law of exponents:

$6a^xa^yaa^2=6a^{x+y+2+1}=6a^{x+y+3}$

Therefore the correct answer is d.

Important note:

Here we need to emphasize that we should always ask the question - what is the exponent being applied to?

For example, in this problem the exponent applies only to the bases of-

$a$$$and not to the numbers, more clearly, in the following expression: $5c^7$the exponent applies only to $c$ and not to the number 5,

whereas when we write:$(5c)^7$the exponent applies to each term of the multiplication inside the parentheses,

meaning:$(5c)^7=5^7c^7$

This is actually the application of the law of exponents:

$(w\cdot r)^n=w^n\cdot r^n$

which follows both from the meaning of parentheses and from the definition of exponents.