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

First, let's do some cosmetic work and arrange the problem into a familiar form. We'll 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. We'll also remember that to get the reciprocal of a simple fraction we just 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:

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

rom here the solution becomes clear, we'll continue treating each of the fractions we got between which multiplication is performed,

Notice that in both fractions there are terms in the numerator and denominator with identical bases, so we'll use 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:

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

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

Next, we'll notice that we want to calculate multiplication between two terms with the same base, so we'll use 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:

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

We got the most simplified expression, so we're done.

Therefore the correct answer is C.