Question

The modulus of the product of all the values of \( (2+3i)^{3/5} \) is:

• A. \( \sqrt{2197} \)
• B. \( \sqrt{2245} \)
• C. \( \sqrt{135} \)
• D. \( \sqrt{489} \)

Hint:

For \( z^{p/q} \), the modulus of the product of all \( q \) roots is simply \( |z|^p \).

Answer 1

The Correct Option is A

Step 1: Understand the question.
The expression $(2+3i)^{3/5}$ has exactly $5$ distinct values (because of the fifth root). We want the modulus of the product of all $5$ of them.
Step 2: Find the modulus of the base.
$|2+3i| = \sqrt{2^2 + 3^2} = \sqrt{13}$.
Step 3: Modulus of each single value.
Every one of the five values has the same modulus, namely $|2+3i|^{3/5} = (\sqrt{13})^{3/5}$, since modulus is unaffected by the angle choice.
Step 4: Multiply the moduli.
The modulus of a product is the product of moduli, so the product of all five has modulus \[ \big((\sqrt{13})^{3/5}\big)^{5} = (\sqrt{13})^{3}. \]
Step 5: Simplify.
$(\sqrt{13})^3 = 13\sqrt{13} = \sqrt{13^2 \cdot 13} = \sqrt{2197}$.
Step 6: Match the option.
This is $\sqrt{2197}$, which is option (A).
\[ \boxed{\sqrt{2197}} \]