Simplify 3b^3 × b^7 × b^8 × b^10 + 30b^14: Exponent Practice

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

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

Manage the numbers and algebraic expressions separately (i.e. - the letters), noting that all algebraic multiplication terms have the same base. Therefore we'll apply 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 shown below is a multiplication of three terms with identical bases:

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

From here on we will no longer indicate the multiplication sign, instead simply placing the terms next to each other signifies multiplication.

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 insert the multiplication term with the result that we obtained 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. Taking out the greatest common factor. For the numbers - the greatest common factor of 30 and 3, is 3, and for the letters, the greatest common factor we can take out is:

$b^{14}$

Therefore the factorization that we obtain is as follows:

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

We applied the power law for multiplication between terms with identical bases as shown below

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

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

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

Therefore the correct answer is B

Note:

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