Simplify (1-2i)(1-i)(1+i)(1+2i).
Thank you!
(1-2i)(1+2i) = 1^2+2^2 = 5
(1-i)(1+i) = 1^2+1^2 = 2
5*2 = 10
Thank you Steve!
To simplify the expression (1-2i)(1-i)(1+i)(1+2i), we can multiply the complex numbers together using the distributive property and the fact that i^2 = -1.
Let's break it down step by step:
Step 1: Multiply (1-2i)(1-i) first.
Using the FOIL method (First, Outer, Inner, Last), we have:
(1-2i)(1-i) = 1 - i - 2i + 2i^2
Simplifying this expression:
= 1 - i - 2i + 2(-1)
= 1 - i - 2i - 2
= -1 - 3i
Step 2: Now, multiply (-1 - 3i)(1+i).
Again, using the FOIL method:
(-1 - 3i)(1+i) = -1 - i - 3i - 3i^2
Simplifying this expression:
= -1 - i - 3i - 3(-1)
= -1 - i - 3i + 3
= 2 - 4i
Step 3: Finally, multiply (2 - 4i)(1 + 2i).
Performing the multiplication:
(2 - 4i)(1 + 2i) = 2 + 4i - 8i - 16i^2
Simplifying this expression:
= 2 + 4i - 8i - 16(-1)
= 2 + 4i - 8i + 16
= 18 - 4i
Therefore, the simplified expression is 18 - 4i.