Question

select the correct answer. which expression is equivalent to the given expression? assume the denominator does not equal zero. ($\frac{3c^{2}d^{4}}{2c^{5}d^{3}}$)^3

Explanation:

Step1: Apply power - of - a - quotient rule

$(\frac{3c^{2}d^{4}}{2c^{5}d^{3}})^{3}=\frac{(3c^{2}d^{4})^{3}}{(2c^{5}d^{3})^{3}}$

Step2: Apply power - of - a - product rule

$\frac{(3c^{2}d^{4})^{3}}{(2c^{5}d^{3})^{3}}=\frac{3^{3}(c^{2})^{3}(d^{4})^{3}}{2^{3}(c^{5})^{3}(d^{3})^{3}}$

Step3: Simplify exponents using power rule $(a^{m})^{n}=a^{mn}$

$\frac{3^{3}(c^{2})^{3}(d^{4})^{3}}{2^{3}(c^{5})^{3}(d^{3})^{3}}=\frac{27c^{6}d^{12}}{8c^{15}d^{9}}$

Step4: Simplify using quotient rule $\frac{a^{m}}{a^{n}}=a^{m - n}$

$\frac{27c^{6}d^{12}}{8c^{15}d^{9}}=\frac{27d^{12 - 9}}{8c^{15 - 6}}=\frac{27d^{3}}{8c^{9}}$

Answer:

$\frac{27d^{3}}{8c^{9}}$ (not shown in the options provided, but this is the correct simplified form following the steps)