Simplify

p^-4+m p^2m)^-4
---------------
(p^-4)^-m

I was thinking that you multiply exponents here and I came up with 2 possible answers. Any help is appreciated. Thanks

p-16m + 16

p-8m – 4

To simplify the given expression, let's break it down step by step.

First, let's simplify the numerator:

p^-4 + m ( p^2m )^-4

Since there is no operation between p^-4 and m, we can treat them as separate terms and simplify each one separately.

For p^-4, when the exponent is negative, we can rewrite it as 1 divided by p^4:

1 / p^4 + m ( p^2m )^-4

Now let's focus on the term in parentheses. Similar to p^-4, we can rewrite (p^2m)^-4 as 1 divided by (p^2m)^4:

1 / p^4 + m / ( p^2m )^4

Next, we can simplify ( p^2m )^4 as p^(2*4)m^(1*4):

1 / p^4 + m / p^8m^4

Now, let's simplify the denominator:

(p^-4)^-m

For (p^-4)^-m, we can apply the exponent properties. When we raise a power to another power, we multiply the exponents:

p^4m

Now, let's rewrite the entire expression using the simplified terms:

( 1 / p^4 + m / p^8m^4 ) / p^4m

To simplify further, we need to get a common denominator for the fractions in the numerator. The common denominator is p^8m^4:

[(1 * p^8m^4) + (m * p^4m)] / p^8m^4

Simplifying the numerator:

(p^8m^4 + mp^5m) / p^8m^4

Now, we combine like terms by adding the exponents of p and m:

p^8m^4 + p^5m^2 / p^8m^4

Since the exponents of p and m are different in the terms of the numerator, we cannot combine them any further. Therefore, the simplified expression is:

(p^8m^4 + p^5m^2) / p^8m^4

I hope this explanation helped you understand how to simplify the expression!