Question

The modulus of the square root of the complex number $6 + 8i$ (where $i = \sqrt{-1}$) is ______.

• A. $\sqrt{5}$
• B. $2\sqrt{5}$
• C. $\sqrt{2} \cdot \sqrt{5}$
• D. $2\sqrt{10}$

Hint:

Never waste time calculating the actual complex square roots $(x+iy)^2 = a+ib$ if the question only asks for the MODULUS! Use the golden rule: $|z^n| = |z|^n$, even for fractional powers like $n=1/2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The modulus of the square root of a complex number $z$ is the square root of the modulus of $z$. Formula: $|\sqrt{z}| = \sqrt{|z|}$.

Step 2: Formula Application:
$|z| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.

Step 3: Explanation:
The modulus of the square root is $\sqrt{|z|} = \sqrt{10}$. Looking at the options, $\sqrt{2} \cdot \sqrt{5} = \sqrt{10}$.

Step 4: Final Answer:
The modulus is $\sqrt{2} \cdot \sqrt{5}$.