Let's solve the expression step-by-step to determine its true simplified form.
Given the expression:
[tex]\[ (2 + 2i) - (1 - i) \][/tex]
Step 1: Distribute the negative sign to the terms inside the parentheses:
[tex]\[ 2 + 2i - 1 + i \][/tex]
Step 2: Combine like terms (real parts together and imaginary parts together):
[tex]\[ (2 - 1) + (2i + i) \][/tex]
Step 3: Simplify the expression:
[tex]\[ 1 + 3i \][/tex]
So, the simplified form of the expression [tex]\((2 + 2i) - (1 - i)\)[/tex] is [tex]\(1 + 3i\)[/tex].
Now, let's analyze the statements:
1. The real part of the result is 1.
2. The real part of the result is 3.
3. The imaginary part of the result is 3.
4. The imaginary part of the result is 1.
5. The result is a purely real number.
6. The result is a purely imaginary number.
From our simplified form [tex]\(1 + 3i\)[/tex]:
- The real part is indeed 1.
- The imaginary part is 3.
- The result is neither purely real nor purely imaginary, as it contains both real and imaginary parts.
Hence, the true statements are:
1. The real part of the result is 1.
3. The imaginary part of the result is 3.
