Step 1: Understanding the Concept:
We must perform a sequence of operations on a given complex number: find its conjugate, find the square root of that conjugate, and finally find the modulus (magnitude) of the result. However, using properties of complex moduli, we can drastically simplify the calculation without ever actually calculating the square root explicitly.
Step 2: Key Formula or Approach:
1. Conjugate of $z = x + iy$ is $\bar{z} = x - iy$.
2. Modulus of $z = x + iy$ is $|z| = \sqrt{x^2 + y^2}$.
3. Important property relating modulus and conjugate: $|\bar{z}| = |z|$.
4. Important property relating modulus and powers/roots: $|z^n| = |z|^n$ and $|\sqrt{z}| = \sqrt{|z|}$.
Step 3: Detailed Explanation:
Let the initial complex number be $z = -7 + 24i$ (since $\sqrt{-1} = i$).
The question asks for the modulus of the square root of its conjugate. Mathematically, this is written as:
\[ \text{Required Value} = |\sqrt{\bar{z}}| \]
Using the property that the modulus of a square root is the square root of the modulus:
\[ |\sqrt{\bar{z}}| = \sqrt{|\bar{z}|} \]
Using the property that a complex number and its conjugate have identical magnitudes ($|\bar{z}| = |z|$):
\[ \sqrt{|\bar{z}|} = \sqrt{|z|} \]
So, we simply need to find the square root of the magnitude of our original number $z$.
First, calculate the magnitude $|z|$:
\[ |z| = \sqrt{(-7)^2 + (24)^2} \]
\[ |z| = \sqrt{49 + 576} \]
\[ |z| = \sqrt{625} \]
\[ |z| = 25 \]
Now, take the square root of this magnitude:
\[ \text{Required Value} = \sqrt{|z|} = \sqrt{25} = 5 \]
Step 4: Final Answer:
The value is 5.