Question

simplify: (5c(3c^{2})^{3})
a. (45c^{6})
b. (135c^{6})
c. (45c^{7})
d. (135c^{7})

Explanation:

Step1: Simplify the power of the term in parentheses

First, we deal with \((3c^{2})^{3}\). According to the power - of - a - product rule \((ab)^{n}=a^{n}b^{n}\) and the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), we have:
\((3c^{2})^{3}=3^{3}\times(c^{2})^{3}\)
\(3^{3}=27\) and \((c^{2})^{3}=c^{2\times3}=c^{6}\), so \((3c^{2})^{3}=27c^{6}\)

Step2: Multiply with the remaining term

Now we multiply \(5c\) with \(27c^{6}\). According to the rule of multiplying monomials \(a^{m}\times a^{n}=a^{m + n}\) (when the bases are the same) and the rule of multiplying coefficients:
\(5c\times27c^{6}=(5\times27)\times(c\times c^{6})\)
\(5\times27 = 135\) and \(c\times c^{6}=c^{1 + 6}=c^{7}\)
So \(5c\times(3c^{2})^{3}=135c^{7}\)

Answer:

D. \(135c^{7}\)