Let's simplify the given expression step by step:
The expression to simplify is:
[tex]\[
\left(2 m^4 n^3\right)^{-4} \left(2 m^2 n\right)^7
\][/tex]
### Step 1: Simplify each term separately
#### Simplify [tex]\(\left(2 m^4 n^3\right)^{-4}\)[/tex]:
1. Distribute the exponent [tex]\(-4\)[/tex] to each factor inside the parentheses:
[tex]\[
(2 m^4 n^3)^{-4} = 2^{-4} (m^4)^{-4} (n^3)^{-4}
\][/tex]
2. Simplify each part individually:
[tex]\[
2^{-4} = \frac{1}{2^4} = \frac{1}{16}
\][/tex]
[tex]\[
(m^4)^{-4} = m^{4 \cdot -4} = m^{-16}
\][/tex]
[tex]\[
(n^3)^{-4} = n^{3 \cdot -4} = n^{-12}
\][/tex]
So, we have:
[tex]\[
(2 m^4 n^3)^{-4} = \frac{1}{16} m^{-16} n^{-12}
\][/tex]
#### Simplify [tex]\(\left(2 m^2 n\right)^7\)[/tex]:
1. Distribute the exponent [tex]\(7\)[/tex] to each factor inside the parentheses:
[tex]\[
(2 m^2 n)^7 = 2^7 (m^2)^7 (n)^7
\][/tex]
2. Simplify each part individually:
[tex]\[
2^7 = 128
\][/tex]
[tex]\[
(m^2)^7 = m^{2 \cdot 7} = m^{14}
\][/tex]
[tex]\[
(n)^7 = n^7
\][/tex]
So, we have:
[tex]\[
(2 m^2 n)^7 = 128 m^{14} n^7
\][/tex]
### Step 2: Combine the simplified terms
Combine the results from the two parts we simplified:
[tex]\[
\left(\frac{1}{16} m^{-16} n^{-12}\right) \left(128 m^{14} n^7\right)
\][/tex]
#### Combine the coefficients:
[tex]\[
\frac{1}{16} \times 128 = 8
\][/tex]
#### Combine the [tex]\(m\)[/tex] terms:
[tex]\[
m^{-16} \times m^{14} = m^{-16 + 14} = m^{-2}
\][/tex]
#### Combine the [tex]\(n\)[/tex] terms:
[tex]\[
n^{-12} \times n^7 = n^{-12 + 7} = n^{-5}
\][/tex]
### Step 3: Write the final simplified expression
Combining all parts, we have:
[tex]\[
8 m^{-2} n^{-5}
\][/tex]
This can be written as:
[tex]\[
\frac{8}{m^2 n^5}
\][/tex]
So the simplified expression is:
[tex]\[
\frac{8}{m^2 n^5}
\][/tex]
In the form [tex]\(\frac{[?]}{\left.m^{[]} n\right]}\)[/tex], we get:
[tex]\[
\frac{8}{m^2 n^5}
\][/tex]
