Question

If $(3 + 4i)^{2025} = 5^{2023}(x + iy)$, then find $\sqrt{x^2 + y^2}$.

• A. 5
• B. 25
• C. 125
• D. 625

Hint:

Use modulus properties of complex numbers and powers: $|z^n| = |z|^n$ to simplify such problems.

Answer 1

The Correct Option is B

Given: \[ (3 + 4i)^{2025} = 5^{2023} (x + iy). \] First, find the modulus of $3 + 4i$: \[ |3 + 4i| = \sqrt{3^2 + 4^2} = 5. \] Therefore, \[ |(3 + 4i)^{2025}| = |3 + 4i|^{2025} = 5^{2025}. \] From the equation: \[ 5^{2025} = 5^{2023} |x + iy| \implies |x + iy| = \frac{5^{2025}}{5^{2023}} = 5^2 = 25. \] Hence, \[ \sqrt{x^2 + y^2} = 25. \] So, the correct answer is 25.