How do you simplify #(-10)^5/ (-10)^9#?

#x^a/x^b=x^(a-b)#

When, in a fraction, the same quantity appears as a multiplicative factor in both numerator and denominator, it can be simplified.

By multiplicative factor, I mean that you can simplify the #2#'s here:

#\frac{cancel(2)x}{cancel(2)y} = \frac{x}{y}#

but not here:

#\frac{x+2}{y+2} \ne \frac{x}{y}#

In your case, the factors stand alone, so they can be simplified. Just remember the very definition of power as reiterated multiplication to write (I'm setting #x=-10# just for aestethic reasons) #\frac{x^5}{x^9}# as

#\frac{x\cdotx\cdotx\cdotx\cdotx}{x\cdotx\cdotx\cdotx\cdotx\cdotx\cdotx\cdotx\cdotx}#

As you can see, the same quantity #x# appears a lot of times in both numerator and denominator, and so it can be simplified:

#\frac{\cancel(x)\cdot\cancel(x)\cdot\cancel(x)\cdot\cancel(x)\cdot\cancel(x)}{\cancel(x)\cdot\cancel(x)\cdot\cancel(x)\cdot\cancel(x)\cdot\cancel(x)\cdotx\cdotx\cdotx\cdotx}=\frac{1}{x^4}#

So, the answer is

#\frac{1}{(-10)^4}=\frac{1}{10000}#

N.B.: in general, when you have the same quantity appearing in both numerator and denominator, you can simply perform some exponent algebra to get

#\frac{x^a}{x^b}=x^{a-b}#

the reason is exactly the reiterated multiplication cancelation that I just showed you. In this example, in fact, you had #\frac{(-10)^5}{(-10)^9} = (-10)^{5-9} = (-10)^{-4}#.

Negative exponent means to consider the inverse of the positive exponent, and in fact we have

#(-10)^{-4} = \frac{1}{10^4}=\frac{1}{10000}#.

Question: Simplify #(-10)^5/(-10)^9#

Using the index rule that #a^n/a^m = a^(n-m)#, we can see that here a = -10, n = 5, and m = 9.

Our expression is now #(-10)^(5-9) = (-10)^-4#.

The next parts don't necessarily simplify the answer, but they make it a bit easier to visualise.

A negative power means we put one over our answer; think of it as a special case of the index rule above, but where n = 0.

Our expression is now #1/(-10)^4# and we're almost done.

We know that a number to an even power can be even or odd (like #2^2 = (-2)^2 = 4#), so in this case we can write out answer as #1/10^4#. Writing out all the zeros, this is #1/(10,000)#.