Simplify b^22/b^20 × b^30/b^20: Variable Power Multiplication

Simplify the following problem:

$\frac{b^{22}}{b^{20}}\times\frac{b^{30}}{b^{20}}=$

Let's start with multiplying the fractions, remembering that the multiplication of fractions is performed by multiplying the numerator by numerator and the denominator by the denominator:

$\frac{b^{22}}{b^{20}}\cdot\frac{b^{30}}{b^{20}}=\frac{b^{22}\cdot b^{30}}{b^{20}\cdot b^{20}}$

In both the numerator and denominator, multiplication occurs between terms with identical bases, so we'll apply the power law for multiplying terms with identical bases:

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

This law can only be used when multiplication is performed between terms with identical bases.

From here on, we will no longer indicate the multiplication sign, instead we will place terms next to each other.
Let's return to the problem and apply the above power law separately to the fraction's numerator and denominator:

$\frac{b^{22}b^{30}}{b^{20}b^{20}}=\frac{b^{22+30}}{b^{20+20}}=\frac{b^{52}}{b^{40}}$

In the final step we calculated the sum of the exponents in the numerator and denominator.

Note that division is required between two terms with identical bases, hence we'll apply the power law for division between terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$

This law can only be used when division is performed between terms with identical bases.

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

$\frac{b^{52}}{b^{40}}=b^{52-40}=b^{12}$

In the final step we calculated the subtraction between the exponents.

This is the most simplified form of the expression:

Therefore, the correct answer is C.