Question

which expression is equivalent to \\(\frac{c^9(d^3)^3}{c^2}\\) for all values of c for which the expression is defined?

\\(\bigcirc\\ c^7d^9\\)

\\(\bigcirc\\ c^6d^6\\)

\\(\bigcirc\\ c^{18}d^{18}\\)

\\(\bigcirc\\ c^4d^{10}\\)

Explanation:

Step1: Simplify the exponent of \( d \)

Using the power of a power rule \((a^m)^n = a^{mn}\), for \((d^3)^3\), we have \( d^{3\times3}=d^9 \). So the expression becomes \(\frac{c^9d^9}{c^2}\).

Step2: Simplify the exponent of \( c \)

Using the quotient of powers rule \(\frac{a^m}{a^n}=a^{m - n}\), for \(\frac{c^9}{c^2}\), we have \( c^{9 - 2}=c^7 \).

Combining the simplified parts for \( c \) and \( d \), we get \( c^7d^9 \).

Answer:

\( c^7d^9 \) (corresponding to the first option: \( c^7d^9 \))