Sure, let's evaluate [tex]\((\sqrt{3 + 2i})^6\)[/tex] step by step.
### Step 1: Express the complex number in polar form
The complex number given is [tex]\(3 + 2i\)[/tex].
First, we need to find the magnitude (also called the modulus) of the complex number:
[tex]\[ |3 + 2i| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \][/tex]
Next, we need to find the argument (also called the angle) of the complex number. The argument can be determined by:
[tex]\[ \arg(3 + 2i) = \tan^{-1}\left(\frac{2}{3}\right) \][/tex]
So, the polar form of the complex number [tex]\(3 + 2i\)[/tex] is:
[tex]\[ 3 + 2i = \sqrt{13} \left(\cos(\theta) + i \sin(\theta)\right) \][/tex]
where [tex]\(\theta = \tan^{-1}\left(\frac{2}{3}\right)\)[/tex].
### Step 2: Find the square root of the complex number
To find the square root of [tex]\(3 + 2i\)[/tex], we use the formula for the square root of a complex number in polar form:
[tex]\[ \sqrt{r (\cos(\theta) + i \sin(\theta))} = \sqrt{r} \left( \cos\left(\frac{\theta}{2}\right) + i \sin\left(\frac{\theta}{2}\right) \right) \][/tex]
Therefore:
[tex]\[ \sqrt{3 + 2i} = \sqrt[4]{13} \left( \cos\left(\frac{\theta}{2}\right) + i \sin\left(\frac{\theta}{2}\right) \right) \][/tex]
### Step 3: Raise the square root to the sixth power
We need to evaluate [tex]\((\sqrt{3 + 2i})^6\)[/tex]. Using properties of exponents:
[tex]\[ \left(\sqrt{3 + 2i}\right)^6 = \left( \sqrt[4]{13} \left( \cos\left(\frac{\theta}{2}\right) + i \sin\left(\frac{\theta}{2}\right) \right) \right)^6 \][/tex]
Using De Moivre's theorem:
[tex]\[ \left(r (\cos(\phi) + i \sin(\phi))\right)^n = r^n (\cos(n \phi) + i \sin(n \phi)) \][/tex]
The base [tex]\( \sqrt{3 + 2i} \)[/tex] becomes:
[tex]\[ \left(\sqrt[4]{13}\right)^6 = (\sqrt{13})^{3} = 13^{3/2} \][/tex]
[tex]\[ 13^{3/2} (\cos\left(6 \cdot \frac{\theta}{2}\right) + i \sin\left(6 \cdot \frac{\theta}{2}\right)) = 13^{3/2} (\cos(3\theta) + i \sin(3\theta)) \][/tex]
Since the magnitude is the same and the exponential form is preserved, our final result is:
[tex]\[ (\sqrt{3 + 2i})^6 = -9 + 46i \][/tex]
Thus, the sought value of [tex]\((\sqrt{3 + 2i})^6\)[/tex] is:
[tex]\[ \boxed{-9 + 46i} \][/tex]
