Certainly! Let's simplify the given expression step by step:
[tex]\[
\left( 2m^4 n^3 \right)^{-4} \left( 2m^2 n \right)^7
\][/tex]
Step 1: Apply the power rule:
For [tex]\((a^m)^n\)[/tex], the power rule states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
Applying this:
[tex]\[
(2m^4 n^3)^{-4} = 2^{-4} \cdot (m^4)^{-4} \cdot (n^3)^{-4} = 2^{-4} \cdot m^{-16} \cdot n^{-12}
\][/tex]
[tex]\[
(2m^2 n)^7 = 2^7 \cdot (m^2)^7 \cdot n^7 = 2^7 \cdot m^{14} \cdot n^7
\][/tex]
Step 2: Multiply the results:
Now we combine these two expressions:
[tex]\[
(2^{-4} \cdot m^{-16} \cdot n^{-12}) \cdot (2^7 \cdot m^{14} \cdot n^7)
\][/tex]
Step 3: Combine the coefficients and like bases:
For the coefficient [tex]\(2\)[/tex]:
[tex]\[
2^{-4} \cdot 2^7 = 2^{-4+7} = 2^3 = 8
\][/tex]
For [tex]\(m\)[/tex]:
[tex]\[
m^{-16} \cdot m^{14} = m^{-16+14} = m^{-2}
\][/tex]
For [tex]\(n\)[/tex]:
[tex]\[
n^{-12} \cdot n^7 = n^{-12+7} = n^{-5}
\][/tex]
Step 4: Write the final simplified expression:
Combining these, we get:
[tex]\[
8m^{-2} n^{-5} = \frac{8}{m^2 n^5}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{8}{m^2 n^5}
\][/tex]
So, your final answer is:
[tex]\[
\frac{8}{m^2 n^5}
\][/tex]
