Question

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

Answer 1

`y^8/x^2`

Explanation:

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`