Evaluate the following powers of $i$.
${\left(3i\right)}^4$

• A. $81$
• B. $-81$
• C. $9$
• D. $-9$

Answer: (A)

Verified step by step guidance

1. Recognize that the expression to evaluate is \(\left(3i\right)^4\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
2. Use the property of exponents to rewrite the expression as \(\left(3i\right)^4 = 3^4 \cdot i^4\).
3. Calculate \$3^4\( by multiplying 3 by itself four times: \)3 \times 3 \times 3 \times 3$.
4. Evaluate \(i^4\) by using the fact that \(i^2 = -1\), so \(i^4 = \left(i^2\right)^2 = (-1)^2\).
5. Multiply the results from steps 3 and 4 together to find the value of \(\left(3i\right)^4\).