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

May 16, 2018

$\frac{1}{10} ^ 4$ or ${10}^{- 4}$

#### Explanation:

${x}^{a} / {x}^{b} = {x}^{a - b}$

May 16, 2018

$\setminus \frac{1}{10000}$

#### 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:

$\setminus \frac{\cancel{2} x}{\cancel{2} y} = \setminus \frac{x}{y}$

but not here:

$\setminus \frac{x + 2}{y + 2} \setminus \ne \setminus \frac{x}{y}$

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) $\setminus \frac{{x}^{5}}{{x}^{9}}$ as

$\setminus \frac{x \setminus \cdot x \setminus \cdot x \setminus \cdot x \setminus \cdot x}{x \setminus \cdot x \setminus \cdot x \setminus \cdot x \setminus \cdot x \setminus \cdot x \setminus \cdot x \setminus \cdot x \setminus \cdot x}$

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

$\setminus \frac{\setminus \cancel{x} \setminus \cdot \setminus \cancel{x} \setminus \cdot \setminus \cancel{x} \setminus \cdot \setminus \cancel{x} \setminus \cdot \setminus \cancel{x}}{\setminus \cancel{x} \setminus \cdot \setminus \cancel{x} \setminus \cdot \setminus \cancel{x} \setminus \cdot \setminus \cancel{x} \setminus \cdot \setminus \cancel{x} \setminus \cdot x \setminus \cdot x \setminus \cdot x \setminus \cdot x} = \setminus \frac{1}{{x}^{4}}$

$\setminus \frac{1}{{\left(- 10\right)}^{4}} = \setminus \frac{1}{10000}$

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

$\setminus \frac{{x}^{a}}{{x}^{b}} = {x}^{a - b}$

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

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

${\left(- 10\right)}^{- 4} = \setminus \frac{1}{{10}^{4}} = \setminus \frac{1}{10000}$.

May 16, 2018

$\frac{1}{10} ^ 4$

#### Explanation:

Question: Simplify ${\left(- 10\right)}^{5} / {\left(- 10\right)}^{9}$

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 ${\left(- 10\right)}^{5 - 9} = {\left(- 10\right)}^{-} 4$.

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 $\frac{1}{- 10} ^ 4$ and we're almost done.

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