((4c^(3)d^(-1))^(4))/(2^(-1)c^(-2)d^(5))

Apply Power Rule: First, we will simplify the expression by applying the power of a power rule and the negative exponent rule. The power of a power rule states that (a^m)^n = a^(m*n). The negative exponent rule states that a^(-n) = 1/a^n. Let's apply these rules to the given expression. ((4c^(3)d^(-1))^(4))/(2^(-1)c^(-2)d^(5)) = (4^4 * c^(3*4) * d^(-1*4)) / (2^(-1) * c^(-2) * d^(5)) = (256 * c^(12) * d^(-4)) / (1/2 * 1/c^(2) * d^(5))Simplify by Multiplying: Next, we will simplify the expression further by multiplying the terms in the numerator and the denominator. 256 * c^(12) * d^(-4) * 2 * c^(2) * 1/d^(5) = 512 * c^(12+2) * d^(-4-5) = 512 * c^(14) * d^(-9)Move Negative Exponents: Now, we will apply the negative exponent rule again to move the negative exponents from the denominator to the numerator. 512 * c^(14) * d^(-9) = 512 * c^(14) / d^(9)Final Answer: Finally, we have the simplified form of the expression. The final answer is 512 * c^(14) / d^(9).

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Calculus

Find derivatives of using multiple formulae

Full solution

\frac{\left(4 c^{3} d^{-1}\right)^{4}}{2^{-1} c^{-2} d^{5}}

Apply Power Rule: First, we will simplify the expression by applying the power of a power rule and the negative exponent rule. $\newline$ The power of a power rule states that $(a^m)^n = a^{m*n}$ . $\newline$ The negative exponent rule states that $a^{-n} = 1/a^n$ . $\newline$ Let's apply these rules to the given expression. $\newline$ $((4c^{3}d^{-1})^{4})/(2^{-1}c^{-2}d^{5}) = (4^4 \cdot c^{3\cdot4} \cdot d^{-1\cdot4}) / (2^{-1} \cdot c^{-2} \cdot d^{5})$ $\newline$ $= (256 \cdot c^{12} \cdot d^{-4}) / (1/2 \cdot 1/c^{2} \cdot d^{5})$
Simplify by Multiplying: Next, we will simplify the expression further by multiplying the terms in the numerator and the denominator. $\newline$ $256 \cdot c^{12} \cdot d^{-4} \cdot 2 \cdot c^{2} \cdot \frac{1}{d^{5}} = 512 \cdot c^{12+2} \cdot d^{-4-5}$ $\newline$ $= 512 \cdot c^{14} \cdot d^{-9}$
Move Negative Exponents: Now, we will apply the negative exponent rule again to move the negative exponents from the denominator to the numerator. $512 \times c^{14} \times d^{-9} = 512 \times c^{14} / d^{9}$
Final Answer: Finally, we have the simplified form of the expression. $\newline$ The final answer is $512 \times c^{14} / d^{9}$ .