Multiplication of Powers: Factoring Out the Greatest Common Factor (GCF)

Let's first deal with the first term which is the multiplication term:

$3b^3\cdot b^7\cdot b^8\cdot b^{10}$

We'll handle separately the numbers and algebraic expressions (i.e. - the letters), noting that all algebraic multiplication terms have the same base, therefore we'll use the power law for multiplication between terms with identical bases:

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

Note that this law applies to any number of terms in multiplication and not just for two, for example for multiplication of three terms with identical base we get:

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

When we used the above power law twice, we can also perform the same calculation for four terms in multiplication five etc...

From here on we will no longer indicate the multiplication sign, but use the conventional writing form where placing terms next to each other means multiplication.

Let's return to the problem and apply the above power law for the multiplication term above:

$3b^3b^7b^8b^{10}=3b^{3+7+8+10}=3b^{28}$

Where we handled the numbers and letters separately.

Let's return to the original question and substitute the multiplication term with the result we got in the last step:

$3b^3b^7b^8b^{10}+30b^{14}=3b^{28}+30b^{14}$

Note that we can further simplify the expression by using factorization by taking out the greatest common factor, for the numbers - we can take out of the parentheses the greatest common factor of 30 and 3, which is 3, and for the letters, the greatest common factor we can take out is:

$b^{14}$

Therefore the factorization we get is:

$3b^{28}+30b^{14}=3b^{14}(b^{14}+10)$

Where we used the power law for multiplication between terms with identical bases to know that:

$b^{28}=b^{14}\cdot b^{14}$

We got the most simplified and factored expression for the above problem:

$3b^{14}(b^{14}+10)$

Therefore the most correct answer is B

Small note:

You can always expand the parentheses after factorization to verify that the factorization was done correctly.

We will factorize by extracting the largest common factor. In the given expression, there are the following terms:

$$$2a^5,\hspace{4pt}8a^6,\hspace{4pt}4a^3$

We'll start with the letters, using the law of exponents for multiplication between terms with identical bases, but in reverse order:

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

to understand that:

$a^6=a^3\cdot a^3\\ a^5=a^3\cdot a^2$

and we'll note that the highest power of $a$that can be extracted as a common factor for all three terms is the power of 3, meaning: $a^3$,

we'll continue and extract the largest common factor for the numbers 2,4,8, which is clearly the number 2 since it is a prime number, therefore we'll conclude and extract the common factor: $2a^3$

We'll apply this to the given expression and extract a common factor:

$2a^5+8a^6+4a^3 =2a^3(a^2+4a^4+2)$

where after extracting the common factor outside the parentheses, we'll look at each term before extracting the common factor separately, asking the question: "By how much did we multiply the common factor to get the current term?" and we'll fill in the missing parts inside the parentheses while making sure that the sign of the term we completed inside the parentheses when multiplied by the sign of the term we extracted outside the parentheses gives us the sign of the original term, it is recommended to verify that the factorization is correct by opening the parentheses, performing the multiplications and confirming that we indeed get the expression before factorization.

Therefore the correct answer is answer A.

First, we use the power law to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$It is necessary to keep in mind that:

$x^4=x^3\cdot x$Next, we return to the problem and extract the greatest common factor for the numbers separately and for the letters separately,

For the numbers, the greatest common factor is

$4$and for the letters it is:

$x^3$and therefore for the extraction

$4x^3$outside the parenthesis

We obtain the expression:

$4x^3+8x^4=4x^3(1+2x)$To determine what the expression inside the parentheses is, we use the power law, our knowledge of the multiplication table, and the answer to the question: "How many times do we multiply the common factor that we took out of the parenthesis to obtain each of the terms of the original expression that we factored?

Therefore, the correct answer is: a.

It is always recommended to review again and check that you get each and every one of the terms of the expression that is factored when opening the parentheses (through the distributive property), this can be done in the margin, on a piece of scrap paper, or by marking the factor we removed and each and every one of the terms inside the parenthesis, etc.