Let's simplify the given expression step by step:
Given expression is:
[tex]\[
\left(-\frac{m^5}{m^{-1} n^{-3} \cdot n^5 \cdot m^{-1} n^0}\right)^4
\][/tex]
### Step 1: Simplify the denominator
First, we simplify the denominator:
[tex]\[
m^{-1} n^{-3} \cdot n^5 \cdot m^{-1} n^0
\][/tex]
Combine the [tex]\(m\)[/tex] terms and the [tex]\(n\)[/tex] terms separately:
[tex]\[
m^{-1} \cdot m^{-1} = m^{-1-1} = m^{-2}
\][/tex]
[tex]\[
n^{-3} \cdot n^5 \cdot n^0 = n^{-3+5+0} = n^2
\][/tex]
Thus, the denominator simplifies to:
[tex]\[
m^{-2} n^2
\][/tex]
### Step 2: Rewrite the fraction
With the simplified denominator, the given expression becomes:
[tex]\[
-\frac{m^5}{m^{-2} n^2}
\][/tex]
Combine the [tex]\(m\)[/tex] terms in the numerator and denominator:
[tex]\[
-\frac{m^5}{m^{-2} n^2} = -\frac{m^{5 - (-2)}}{n^2} = -\frac{m^{5 + 2}}{n^2} = -\frac{m^7}{n^2}
\][/tex]
### Step 3: Raise to the fourth power
After simplification, we raise the expression [tex]\(-\frac{m^7}{n^2}\)[/tex] to the fourth power:
[tex]\[
\left(-\frac{m^7}{n^2}\right)^4
\][/tex]
Applying the power rule [tex]\((a^b)^c = a^{b \cdot c}\)[/tex], we get:
[tex]\[
\left(-1 \cdot \frac{m^7}{n^2}\right)^4 = (-1)^4 \cdot \left(\frac{m^7}{n^2}\right)^4 = 1 \cdot \frac{(m^7)^4}{(n^2)^4} = \frac{m^{28}}{n^8}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{m^{28}}{n^8}
\][/tex]
### Step 4: Substitution check (if needed)
To verify correctness, we can substitute [tex]\(m = 1\)[/tex] and [tex]\(n = 1\)[/tex]:
[tex]\[
\frac{1^{28}}{1^8} = \frac{1}{1} = 1
\][/tex]
This confirms the result is correctly simplified.
### Final Result
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{m^{28}}{n^8}}
\][/tex]
And when substituting [tex]\(m = 1\)[/tex] and [tex]\(n = 1\)[/tex]:
[tex]\[
\boxed{1}
\][/tex]
