Question

The square root of \( 7 + 24i \) is:

• A. \( 4 - 3i \)
• B. \( 3 + 4i \)
• C. \( 3 - 4i \)
• D. \( 4 + 3i \)

Hint:

For square roots of complex numbers, convert the complex number to polar form and apply the square root formula for complex numbers. Then convert back to rectangular form.

Answer 1

The Correct Option is D

We are asked to find the square root of the complex number \( 7 + 24i \). To find \( \sqrt{7 + 24i} \), we use the fact that the square root of a complex number \( z = x + yi \) can be expressed as \( \sqrt{r}(\cos \theta + i \sin \theta) \), where \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}(\frac{y}{x}) \). Here, \( 7 + 24i \) has \( x = 7 \) and \( y = 24 \). Step 1: Finding the magnitude The magnitude \( r \) is: \[ r = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25. \] Step 2: Finding the argument The argument \( \theta \) is: \[ \theta = \tan^{-1}\left( \frac{24}{7} \right) \approx 74.05^\circ. \] Step 3: Applying the square root formula Using the formula for the square root of a complex number, the square root of \( 7 + 24i \) is: \[ \sqrt{7 + 24i} = \sqrt{25} \left( \cos \frac{74.05^\circ}{2} + i \sin \frac{74.05^\circ}{2} \right). \] After calculating, we find that: \[ \sqrt{7 + 24i} = 4 + 3i. \] Thus, the square root of \( 7 + 24i \) is \( 4 + 3i \).