Multiplication of Powers: Numbers as coefficients

Given that there is a multiplication between all terms in the expression, we will apply the distributive property of multiplication. This allows us to handle the coefficients of terms raised to powers, as well as the terms themselves separately. For added clarity, let's handle it in steps:

$2a^2\cdot3a^4=2\cdot3\cdot a^2\cdot a^4=6\cdot a^2\cdot a^4$

Due to the multiplication 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:$2a^2\cdot3a^4=6\cdot a^2\cdot a^4$

From here on we will no longer write the multiplication sign. We will instead 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 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 the law of exponents for two terms we can equally perform the same calculation for four terms or five and so on..,

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

$6a^2a^4=6a^{2+4}=6a^6$

Therefore the correct answer is C.

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:

$(x\cdot y)^n=x^n\cdot y^n$

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

Note that there is multiplication between all terms in the expression, hence we'll apply the distributive property of multiplication to understand that we can handle the coefficients of terms raised to powers as well as the terms themselves separately. For clarity, let's handle it in steps:

$a^2\cdot2a^3\cdot3a^{-1}=2\cdot3\cdot a^2\cdot a^3\cdot a^{-1}=6\cdot a^2\cdot a^3\cdot a^{-1}$

Given that there is multiplication 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:$a^2\cdot2a^3\cdot3a^{-1}=6\cdot a^2\cdot a^3\cdot a^{-1}$

From here on we will no longer write the multiplication sign and remember that it is conventional to simply place the terms next to each other\ place the term next to its coefficient to indicate multiplication between them,

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

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

Note 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 the following:

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

When we used the above law of exponents twice, we can also perform the same calculation for four terms in multiplication five and so on..,

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

$6a^2a^3a^{-1}=6a^{2+3-1}=6a^4$

Therefore the correct answer is C.

Important note:

Here we need to emphasize that we should always ask the question - what does the exponent apply to?

For example, in this problem the exponent applies only to the base of $a$ and not to the numbers, more clearly, in the following expression: $5b^7$The exponent applies only to $b$ and not to the number 5,

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

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

This is actually the application of the law of exponents:

$(x\cdot y)^n=x^n\cdot y^n$

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

Note that there is multiplication between all terms in the expression. Thus we'll first apply the distributive property of multiplication to understand that we can separately handle the coefficients of the terms raised to powers, and the terms themselves separately. For clarity, let's handle it in steps:

$b^3\cdot3b^2\cdot2b^{-2}=3\cdot2\cdot b^3\cdot b^2\cdot b^{-2}=6\cdot b^3\cdot b^2\cdot b^{-2}$

Given that there is multiplication 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:$b^3\cdot3b^2\cdot2b^{-2}=6\cdot b^3\cdot b^2\cdot b^{-2}$

From here on we will no longer write the multiplication sign and remember that it is conventional to 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:

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

Note 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 the following:

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

When we used the above law of exponents twice, we can also perform the same calculation for four terms in multiplication, five and so on..

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

$6b^3b^2b^{-2}=6b^{3+2-2}=6b^3$

Therefore, the correct answer is A.

Important note:

Here we need to emphasize that we should always ask the question - what does the exponent apply to?

For example, in this problem the exponent applies only to the $b$ bases 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 of the terms in the multiplication within the parentheses,

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

This is actually the application of the law of exponents:

$(x\cdot y)^n=x^n\cdot y^n$

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

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.

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.