Question

which expression is equivalent to ((6i)^{3})?
a. (216)
b. (-18i)
c. (216i)
d. (-216i)

Explanation:

Step1: Recall the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, for \((6i)^3\), we can write it as \(6^3 \times i^3\).

Step2: Calculate \(6^3\)

\(6^3 = 6\times6\times6 = 216\).

Step3: Recall the properties of the imaginary unit \(i\)

We know that \(i^1 = i\), \(i^2 = - 1\), and \(i^3=i^2\times i\). Substituting \(i^2=-1\) into the expression for \(i^3\), we get \(i^3=-1\times i=-i\).

Step4: Multiply the results from Step2 and Step3

We have \(6^3\times i^3 = 216\times(-i)=-216i\).

Answer:

D. \(-216i\)