((u^(4)v^(-2)*uv^(2))^(4))/(2v^(3))

Simplify Inside Parentheses: First, simplify the expression inside the parentheses before raising it to the fourth power. (u^(4)v^(-2)*uv^(2)) = u^(4+1)*v^(-2+2) = u^5*v^0 = u^5, since any number to the power of 0 is 1.Raise to Fourth Power: Now raise the simplified expression inside the parentheses to the fourth power. (u^5)^(4) = u^(5*4) = u^20.Divide by Denominator: Next, divide the result by the denominator 2v^(3). (u^20)/(2v^(3)) = (1/2)*u^20*v^(-3).Final Simplified Form: Finally, write the expression in its simplified form. (1/2)*u^20*v^(-3) is already in its simplest form.

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$25$ ) $\frac{\left(u^{4} v^{-2} \cdot u v^{2}\right)^{4}}{2 v^{3}}$

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25

)

\frac{\left(u^{4} v^{-2} \cdot u v^{2}\right)^{4}}{2 v^{3}}

Simplify Inside Parentheses: First, simplify the expression inside the parentheses before raising it to the fourth power. $\newline$ $(u^{4}v^{-2}*uv^{2}) = u^{4+1}*v^{-2+2} = u^5*v^0 = u^5$ , since any number to the power of $0$ is $1$ .
Raise to Fourth Power: Now raise the simplified expression inside the parentheses to the fourth power. $(u^5)^{4} = u^{5\times4} = u^{20}$ .
Divide by Denominator: Next, divide the result by the denominator $2v^{3}$ . $(u^{20})/(2v^{3}) = (1/2)\cdot u^{20}\cdot v^{-3}.$
Final Simplified Form: Finally, write the expression in its simplified form. $\newline$ $(\frac{1}{2})u^{20}v^{(-3)}$ is already in its simplest form.