Write the following complex numbers in the form `A+iB:(1+2i)^(3)`

Answer

To write the complex number \( (1 + 2i)^3 \) in the form \( A + iB \), we can use the binomial theorem or the identity for the cube of a binomial. Here’s a step-by-step solution:

### Step 1: Identify the components
We have the complex number \( 1 + 2i \). Here, \( a = 1 \) and \( b = 2i \).

### Step 2: Apply the binomial expansion
Using the identity for the cube of a binomial, \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \):
\[ (1 + 2i)^3 = 1^3 + (2i)^3 + 3 \cdot 1 \cdot (2i)(1 + 2i) \]

### Step 3: Calculate \( 1^3 \) and \( (2i)^3 \)
\[ 1^3 = 1 \]
\[ (2i)^3 = 2^3 \cdot i^3 = 8 \cdot (-i) = -8i \]

### Step 4: Calculate \( 3 \cdot 1 \cdot (2i)(1 + 2i) \)
First, calculate \( 2i(1 + 2i) \):
\[ 2i(1 + 2i) = 2i + 4i^2 = 2i + 4(-1) = 2i - 4 = -4 + 2i \]
Now, multiply by 3:
\[ 3 \cdot (-4 + 2i) = -12 + 6i \]

### Step 5: Combine all parts
Now, combine all the parts:
\[ (1 + 2i)^3 = 1 - 8i - 12 + 6i \]
Combine the real and imaginary parts:
\[ = (1 - 12) + (-8i + 6i) = -11 - 2i \]

### Final Result
Thus, the complex number \( (1 + 2i)^3 \) in the form \( A + iB \) is:
\[ -11 - 2i \]
Where \( A = -11 \) and \( B = -2 \).