Question

The modulus of the complex number \( (2\sqrt{2} + i2\sqrt{2})^2 \) is equal to

• A. \(64 \)
• B. \(4 \)
• C. \(32 \)
• D. \(8 \)
• E. \(16 \)

Hint:

Always simplify modulus first, then apply exponent rule \( |z^n| = |z|^n \).

Answer 1

Answer: (E)

Note: The OCR `(2√2 + 12√2)²` seems to be a typo. Based on the context of complex numbers, the question is interpreted as finding the modulus of `(2√2 + i2√2)²`.
Step 1: Understanding the Concept:
The question asks for the modulus of the square of a complex number. We can use the property of modulus: `|z^n| = |z|^n`.
Step 2: Key Formula or Approach:
For a complex number `z = a + bi`, its modulus is `|z| = \sqrt{a^2 + b^2}`.
The most efficient approach is to find the modulus of the base first and then square it.
Let `z = 2\sqrt{2} + i2\sqrt{2}`. We need to find `|z^2|`, which is equal to `|z|^2`.
Step 3: Detailed Explanation:
Method 1: Using the property `|z^n| = |z|^n` (Recommended)
Let `z = 2\sqrt{2} + i2\sqrt{2}`.
First, find the modulus of `z`:
\[ |z| = \sqrt{(2\sqrt{2})^2 + (2\sqrt{2})^2} \] \[ |z| = \sqrt{(4 \cdot 2) + (4 \cdot 2)} = \sqrt{8 + 8} = \sqrt{16} = 4 \] Now, find the modulus of `z^2`:
\[ |z^2| = |z|^2 = 4^2 = 16 \] Method 2: Squaring the complex number first
First, expand the expression:
\[ (2\sqrt{2} + i2\sqrt{2})^2 = (2\sqrt{2})^2 + 2(2\sqrt{2})(i2\sqrt{2}) + (i2\sqrt{2})^2 \] \[ = 8 + i(2 \cdot 2 \cdot 2 \cdot \sqrt{2} \cdot \sqrt{2}) + i^2(8) \] \[ = 8 + i(16) - 8 = 16i \] Now, find the modulus of `16i`:
\[ |16i| = |0 + 16i| = \sqrt{0^2 + 16^2} = \sqrt{256} = 16 \] Both methods yield the same result.
Step 4: Final Answer:
The modulus of the complex number is 16.