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 Question:
We are given a complex number $z = 6 + 8i$. The question asks for the mathematical modulus (absolute value) of the principal square root of this complex number.

Step 2: Detailed Explanation:
Let the given complex number be $z = 6 + 8i$.
We need to evaluate: $|\sqrt{z}|$.
A fundamental and highly useful property of complex numbers is that the modulus operator completely commutes with exponents and fractional powers.
Mathematically: $|\sqrt{z}| = \sqrt{|z|}$.
Therefore, we do not need to perform the grueling algebraic process of actually finding the complex square root ($\pm(a+bi)$). We just need to find the modulus of the original number $z$ and take the real square root of that!
Calculate the modulus of $z$ ($|z|$):
$|z| = \sqrt{a^2 + b^2}$
$|z| = \sqrt{6^2 + 8^2}$
$|z| = \sqrt{36 + 64}$
$|z| = \sqrt{100} = 10$
Now, apply the property to find the modulus of the square root:
$|\sqrt{z}| = \sqrt{|z|} = \sqrt{10}$
Let's look at the options. Option (c) is written in an unsimplified, factored form:
$\sqrt{2} \cdot \sqrt{5}$
By radical rules, $\sqrt{2} \cdot \sqrt{5} = \sqrt{2 \times 5} = \sqrt{10}$.
This matches our derived result perfectly.

Step 3: Final Answer:
The modulus is $\sqrt{2} \cdot \sqrt{5}$, matching option (c).