simplify:
(p^-4+m p^2m)^-4
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(p^-4)^-m
To simplify the expression, we'll start by using the property of exponents that states (a^b)^c = a^(b * c).
Let's begin with the denominator: (p^-4)^-m. This can be simplified as follows:
(p^-4)^-m = p^((-4)*(-m))
= p^(4m)
Now let's move to the numerator: (p^-4 + m p^2m)^-4. Since we have a sum within parentheses raised to a power, we need to use the binomial theorem or expand it using the distributive property.
First, let's expand p^-4 + m p^2m:
p^-4 + m p^2m = p^-4 + m (p^2m)
= p^-4 + m p^(2m)
Now, we can raise this sum to the power of -4:
(p^-4 + m p^2m)^-4 = (p^-4 + m p^(2m))^-4
However, we can simplify this even further by using the property of exponents that states (a + b)^c = a^c + c(a^(c-1))b + ... + b^c when c is a positive integer.
Therefore, we can write:
(p^-4 + m p^2m)^-4 = (p^-4)^-4 + (-4)(p^-4)^(-4-1)(m p^2m) + ...
Simplifying further, we have:
(p^-4 + m p^2m)^-4 = p^(4*4) + (-4)(p^-4)^(-5)(m p^2m) + ...
Now, substituting the previously simplified denominator (p^-4)^-m = p^(4m), we have:
(p^-4 + m p^2m)^-4 = p^(4*4) + (-4)(p^(4m))^(-5)(m p^2m) + ...
Simplifying the exponents, we get:
(p^-4 + m p^2m)^-4 = p^16 + (-4)(p^(20m))(m p^2m) + ...
Finally, you can simplify further or rewrite the terms in a more suitable format if needed for your specific use.