# How do you simplify each expression using positive exponents (x^-2y^-4x^3)^-2 ?

Jan 7, 2016

${y}^{8} / {x}^{2}$

#### Explanation:

We can use exponent rules to simplify this expression. Taking a look at the original function;

${\left({x}^{-} 2 {y}^{-} 4 {x}^{3}\right)}^{-} 2$

We can see that there are two $x$ terms inside of the parenthesis. Lets combine those first. If we multiply two terms with exponents, the exponents add, in other words;

${x}^{a} \times {x}^{b} = {x}^{a + b}$

Applying this to our case, we get;

${\left({x}^{\left(3 - 2\right)} {y}^{-} 4\right)}^{-} 2$

${\left({x}^{1} {y}^{-} 4\right)}^{-} 2$

Now lets take a look at the exponent outside the parenthesis. Whenever we raise an exponent term to an exponent, we multiply the exponents.

${\left({x}^{a}\right)}^{b} = {x}^{\left(a\right) \left(b\right)}$

In our case, raising both the $x$ term and the $y$ term to $- 2$ we get;

${x}^{\left(- 2\right) \left(1\right)} {y}^{\left(- 2\right) \left(- 4\right)}$

${x}^{-} 2 {y}^{8}$

Now we have one negative exponent and one positive exponent. We need to convert the $x$ term to a positive exponent. To do that we will invert the term.

${x}^{-} a = \frac{1}{x} ^ a$

So to get rid of the $-$ we will move the $x$ term to the denominator.

${y}^{8} / {x}^{2}$