To simplify the expression

$(m^2)^3 (mn)^{-1} (n^3)^{\frac{2}{3}}$

we proceed step by step using the laws of exponents. The laws we will use are:

1. $(a^m)^n = a^{m \cdot n}$,
2. $a^m \cdot a^n = a^{m + n}$,
3. $\frac{a^m}{a^n} = a^{m - n}$,
4. $a^{-n} = \frac{1}{a^n}$.

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Step 1: Simplify each component

1. $(m^2)^3 = m^{2 \cdot 3} = m^6$,
2. $(mn)^{-1} = m^{-1}n^{-1}$,
3. $(n^3)^{\frac{2}{3}} = n^{3 \cdot \frac{2}{3}} = n^2$.

So the expression becomes: $m^6 \cdot m^{-1}n^{-1} \cdot n^2.$

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Step 2: Combine like terms

• For $m$: $m^6 \cdot m^{-1} = m^{6 - 1} = m^5$,
• For $n$: $n^{-1} \cdot n^2 = n^{-1 + 2} = n^1 = n$.

Thus, the simplified expression is: $m^5n.$

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Final Answer:

$\boxed{m^5n}$

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Do you want details on any part of the solution? Let me know!

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Related Questions:

1. How do the laws of exponents simplify operations with negative exponents?
2. What does $(a^b)^c = a^{bc}$ mean in practical use?
3. How do fractional exponents relate to roots?
4. Can this process handle variables with unknown coefficients?
5. How does this apply to expressions with multiple variables and terms?

Tip: Always simplify step-by-step and group like terms systematically to avoid mistakes!