-3(2 3i)(1-i)⁴ 5(-2 i)(1 3i)²-2(-3 4i)=?

Question

Answer

Solution:

We can simplify the given expression using the following steps:

Step 1: Expand the terms

Using the distributive property, we can expand the terms in the expression:

-3(2 + 3i)(1 - i)⁴ + 5(-2i)(1 + 3i)² - 2(-3 + 4i)

Expanding the first term:

-3(2 + 3i)(1 - i)⁴ = -3(2 + 3i)(1 - 4i + 6i² - 4i³ + i⁴)

Since i² = -1 and i³ = -i and i⁴ = 1, we can simplify the above expression:

= -3(2 + 3i)(1 - 4i - 6 + 4i + 1)

= -3(2 + 3i)(-4i - 5)

= 6(4i + 5 + 6i - 8i² - 10i - 15i²)

= 6(18 - 6i)

= 108 - 36i

Expanding the second term:

5(-2i)(1 + 3i)² = 5(-2i)(1 + 6i + 9i²)

Since i² = -1, we can simplify the above expression:

= 5(-2i)(1 + 6i - 9)

= -5(18i + 2i)

= -100i

Expanding the third term:

-2(-3 + 4i) = 6 - 8i

Step 2: Combine like terms

Now we can combine the simplified terms:

108 - 36i - 100i + 6 - 8i = 114 - 144i

Step 3: Final answer

Therefore, the final answer is 114 - 144i.