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

##y^8/x^2##

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

##(x^-2y^-4x^3)^-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 xx x^b = x^(a+b)##

Applying this to our case, we get;

##(x^((3-2))y^-4)^-2##

##(x^1y^-4)^-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.

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

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

##x^((-2)(1))y^((-2)(-4))##

##x^-2y^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 = 1/x^a##

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

##y^8/x^2##