Simplify the Expression: (b^10/b^2)÷(b^9/b^5)

Simplify the following problem:

$\frac{b^{10}}{b^2}:\frac{b^9}{b^5}=$

Let's begin by rearranging the problem into a more workable format. Write the expression in an organized way using fractions, remembering that division is actually multiplication by the reciprocal, and therefore instead of dividing by a fraction we can always multiply by its reciprocal. In order to obtain the reciprocal of a simple fraction we simply flip between the numerator and denominator. Mathematically, instead of writing:$:\frac{x}{y}$We can always write:$\cdot\frac{y}{x}$

Let's apply this to the problem:

$\frac{b^{10}}{b^2}:\frac{b^9}{b^5}=\frac{b^{10}}{b^2}\cdot\frac{b^5}{b^9}$

From here the solution becomes clear, we'll continue to multiply the fractions together.

Notice that in both fractions there are terms in the numerator and denominator with identical bases, hence we'll apply the division law for terms with identical bases to simplify the expression:

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

Let's apply this law to each fraction separately:

$\frac{b^{10}}{b^2}\cdot\frac{b^5}{b^9}=b^{10-2}\cdot b^{5-9}=b^8\cdot b^{-4}$

In the second stage we calculated the result of the subtraction operation in the exponents for each term separately,

The next step is to calculate the multiplication operation between two terms with the same base, hence we'll apply the power law for multiplication between terms with identical bases:

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

Let's apply this law to the problem:

$b^8\cdot b^{-4}=b^{8+(-4)}=b^{8-4}=b^4$

The expression is now in its most simplified form.

Therefore the correct answer is C.