Question

Perform the indicated operations. Write your answer in your notebook.
(2i²−3i³+2i⁴−2i⁵)⁶

Answer 1

The given complex expression simplifies to i, and raising i to the sixth power gives us -1 as the final solution.

Step-by-step explanation:

To simplify the expression (2i²−3i³+2i´−2iµ)⁶, we first need to understand that i is the imaginary unit with the property that i² = -1. Let's simplify the base of the expression using this property:

• i² = -1
• i³ = i² · i = -1 · i = -i
• i´ = (i²)² = (-1)² = 1
• iµ = i´ · i = 1 · i = i

Now, substituting these values into the original expression:

• 2i² becomes 2(-1) = -2
• -3i³ becomes -3(-i) = 3i
• 2i´ becomes 2(1) = 2
• -2iµ becomes -2(i) = -2i

Combining like terms, we get:

• -2 + 3i + 2 - 2i = -2 + 2 + 3i - 2i = 0 + i = i

So, the expression simplifies to i. Now we raise i to the power of 6: i⁶. Since i´ = 1 and i² = -1, we can write i⁶ as (i´)(i²) = 1(-1) = -1. Therefore, the given expression simplifies to -1.