Simplify the following expression.

i^7(12/6i^3)

To simplify the given expression \( i^7 \left(\frac{12}{6i^3}\right) \), follow these steps:

1. Simplify the fraction \(\frac{12}{6i^3}\):
   \[
   \frac{12}{6i^3} = 2i^{-3}
   \]

2. Now, multiply this result by \(i^7\):
   \[
   i^7 \cdot 2i^{-3}
   \]

3. Combine the powers of \(i\):
   \[
   2i^{7 - 3} = 2i^4
   \]

4. Recall the powers of \(i\):
   \[
   i^4 = (i^2)^2 = (-1)^2 = 1
   \]

5. Substitute \(i^4\) with 1:
   \[
   2i^4 = 2 \cdot 1 = 2
   \]

Therefore, the simplified expression is:
\[
2
\]

Would you like more details or have any questions? Here are some related questions:

1. What are the properties of the imaginary unit \(i\)?
2. How do you simplify expressions involving imaginary numbers?
3. What are the steps to simplify complex fractions?
4. How do the powers of \(i\) cycle?
5. What is the significance of \(i^4 = 1\)?
6. Can you provide examples of simplifying other expressions with imaginary numbers?
7. How can you verify the simplification of an expression involving \(i\)?
8. What are some common mistakes when working with powers of \(i\)?

**Tip:** Remember that \(i\) cycles every four powers: \(i, -1, -i, 1\).

Learn how to simplify the expression i^7(12/6i^3) involving imaginary numbers with a detailed step-by-step solution. This problem covers complex number concepts and the properties of the imaginary unit (i). Suitable for students in Grades 10-12.

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