Question

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

Answer 1

`1/10^4` or `10^(-4)`

Explanation:

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

Answer 2

`\frac{1}{10000}`

Explanation:

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}`.

Answer 3

`1/10^4`

Explanation:

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)`.