Finding the Argument of Complex Number Power

The question asks us to find the argument of the complex number $\left(\frac{2+3i}{12-4i}\right)^4$. We need to follow a step-by-step process to simplify the expression and then find its argument.

Step 1: Simplify the Complex Fraction

First, let's simplify the complex number inside the parenthesis, which is $z = \frac{2+3i}{12-4i}$. To do this, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $12-4i$ is $12+4i$.

$$ z = \frac{2+3i}{12-4i} \times \frac{12+4i}{12+4i} $$

Now, we multiply the numerators:

Numerator: $$(2+3i)(12+4i) = (2 \times 12) + (2 \times 4i) + (3i \times 12) + (3i \times 4i)$$

$$ = 24 + 8i + 36i + 12i^2 $$

Since $i^2 = -1$, we have:

$$ = 24 + 44i + 12(-1) = 24 + 44i - 12 = 12 + 44i $$

Next, we multiply the denominators:

Denominator: $$(12-4i)(12+4i) = (12)^2 - (4i)^2$$

$$ = 144 - (16 \times i^2) = 144 - 16(-1) = 144 + 16 = 160 $$

So, the simplified complex number is:

$$ z = \frac{12 + 44i}{160} $$

We can split this into real and imaginary parts:

$$ z = \frac{12}{160} + \frac{44}{160}i $$

Simplifying the fractions:

$$ \frac{12}{160} = \frac{3 \times 4}{40 \times 4} = \frac{3}{40} $$

$$ \frac{44}{160} = \frac{11 \times 4}{40 \times 4} = \frac{11}{40} $$

Therefore, the simplified complex number is $$ z = \frac{3}{40} + \frac{11}{40}i $$

Step 2: Calculate the Argument of the Simplified Number

The complex number is $z = x + yi$, where the real part is $x = \frac{3}{40}$ and the imaginary part is $y = \frac{11}{40}$.

The argument of a complex number $z = x + yi$, denoted as $\arg(z)$, is typically calculated using the arctangent function. When the real part $x$ is positive, the formula is:

$$ \arg(z) = \tan^{-1}\left(\frac{y}{x}\right) $$

Substituting the values of $x$ and $y$:

$$ \arg(z) = \tan^{-1}\left(\frac{11/40}{3/40}\right) $$

$$ \arg(z) = \tan^{-1}\left(\frac{11}{3}\right) $$

Step 3: Apply the Power Rule for Arguments

We need to find the argument of $z^4$. A property of arguments states that for any complex number $z$ and integer $n$, the argument of $z^n$ is $n$ times the argument of $z$. Mathematically:

$$ \arg(z^n) = n \times \arg(z) $$

In this case, $n=4$. So, we have:

$$ \arg\left(z^4\right) = 4 \times \arg(z) $$

Substituting the argument we found in Step 2:

$$ \arg\left(z^4\right) = 4 \times \tan^{-1}\left(\frac{11}{3}\right) $$

This result matches one of the given options.