Question

One of the values of $\sqrt{24-70i} + \sqrt{-24+70i}$ is

• A. $2+12i$
• B. $12-2i$
• C. $-12+2i$
• D. $-12-2i$

Hint:

A quick method to find the square root of $a+ib$ is to use the formula: $\sqrt{a+ib} = \pm \left( \sqrt{\frac{\sqrt{a^2+b^2}+a}{2}} + i \cdot \text{sgn}(b) \sqrt{\frac{\sqrt{a^2+b^2}-a}{2}} \right)$. For $24-70i$, $|z|=74, a=24, b=-70$. $\sqrt{24-70i} = \pm \left( \sqrt{\frac{74+24}{2}} - i \sqrt{\frac{74-24}{2}} \right) = \pm(\sqrt{49} - i\sqrt{25}) = \pm(7-5i)$.

Answer 1

The Correct Option is D

Step 1: Calculate Square Roots of Complex Numbers: Formula: \( \sqrt{a+bi} = \pm \left( \sqrt{\frac{|z|+a}{2}} + (\text{sgn}(b))i \sqrt{\frac{|z|-a}{2}} \right) \). For \( z_1 = 24 - 70i \): \( |z_1| = \sqrt{24^2 + (-70)^2} = \sqrt{576 + 4900} = \sqrt{5476} = 74 \). Since \( b<0 \): \( \sqrt{z_1} = \pm \left( \sqrt{\frac{74+24}{2}} - i \sqrt{\frac{74-24}{2}} \right) = \pm (\sqrt{49} - i\sqrt{25}) = \pm (7 - 5i) \). For \( z_2 = -24 + 70i \): \( |z_2| = 74 \). Since \( b>0 \): \( \sqrt{z_2} = \pm \left( \sqrt{\frac{74-24}{2}} + i \sqrt{\frac{74-(-24)}{2}} \right) = \pm (\sqrt{25} + i\sqrt{49}) = \pm (5 + 7i) \).
Step 2: Find the Sum: We need to find a value of \( \sqrt{z_1} + \sqrt{z_2} \). We can choose signs independently. Possible sums: 1. \( (7-5i) + (5+7i) = 12 + 2i \) (Not in options) 2. \( (7-5i) - (5+7i) = 2 - 12i \) (Not in options) 3. \( -(7-5i) + (5+7i) = -2 + 12i \) (Not in options) 4. \( -(7-5i) - (5+7i) = -7 + 5i - 5 - 7i = -12 - 2i \) Option (D) matches the fourth combination.