Multiplication of Powers: Numbers as coefficients

$x^3\times7x\times2x^{-3}=$

Here we have multiplication between all the elements in the expression, so we will first use the commutative law in multiplication approach the numbers separately, for clarity we will approach it in stages:

$x^3\cdot7x\cdot2x^{-3}=7\cdot2\cdot x^3\cdot x\cdot x^{-3}=14\cdot x^3\cdot x\cdot x^{-3}$Note that it is possible (and even preferable) to skip the middle stage, meaning:

To write directly:$x^3\cdot7x\cdot2x^{-3}=14\cdot x^3\cdot x\cdot x^{-3}$

We will continue and use the associative law for multiplication between elements with the same bases:

$c^m\cdot c^n=c^{m+n}$Note that this law is also valid for several elements in multiplication and not just for two, for example for a multiplication of three elements with the same base we will get:

$c^m\cdot c^n\cdot c^k=c^{m+n}\cdot c^k=c^{m+n+k}$When can use the associative even for four, five, or more elements in a multiplication.

Let's go back to the problem, and apply the associative law:

$14x^3xx^{-3}=14x^{3+1-3}=14x^1=14x$And therefore the correct answer is c.

Important note:

Here it is necessary to emphasize that you always need to ask the question - what do the parentheses apply to?

For example, in the problem here the parentheses only apply to the bases of the-

$x$$$and not to the exponents, in a clearer way, also in the following expression:

$5c^7$The parentheses apply only to $c$and not to the exponent 5, as opposed to that when writing:

$(5c)^7$The parentheses apply to each of the multiplication elements within the parentheses, meaning:

$(5c)^7=5^7c^7$This is actually the application of the associative law:

$(w\cdot r)^n=w^n\cdot r^n$ resulting both from the meaning of the parentheses and from the definition of parentheses.

$14x$

$2a^2\times3a^4=$

$6a^6$

$a^2\times2a^3\times3a^{-1}=$

$6a^4$

$b^3\times3b^2\times2b^{-2}=$

$6b^3$

$a^x\times3a^y\times a^2\times2a=$

$6a^{3+x+y}$