How Do You Simplify This Complex Absolute Value Expression?

[FlufferNuterFSU]

Homework Statement

I need to simplify the expression below. The absolute value is throwing me off

$$\left[\left|(\alpha + k)^{2}e^{-2i \alpha a} - (\alpha - k)^{2}e^{2i \alpha a}\right|\right]^{2}$$

Homework Equations

I know $$\left|e^{ix}\right| = 1$$

The Attempt at a Solution

I know this eventually simplifies to:
$$(\alpha + k)^{4} + (\alpha - k)^{4} - (\alpha^{2} - k^{2})^{2}(e^{4i \alpha a} + e^{-4i \alpha a})$$

 

[FlufferNuterFSU]

Don't worry about it. I figured it out.

 

[Architeuthis Dux]

+ 2(\alpha^{2} + k^{2})(e^{-4i \alpha a} - e^{4i \alpha a})

However, I am having trouble simplifying the expression because of the absolute value. Can you provide any guidance?

To simplify the absolute value in this expression, you can use the property that the absolute value of a product is equal to the product of the absolute values. In this case, you can rewrite the expression as:

\left|(\alpha + k)^{2}e^{-2i \alpha a} - (\alpha - k)^{2}e^{2i \alpha a}\right|^{2} = \left|(\alpha + k)^{2}\right|^{2} \cdot \left|e^{-2i \alpha a}\right|^{2} + \left|(\alpha - k)^{2}\right|^{2} \cdot \left|e^{2i \alpha a}\right|^{2}

Since the absolute value of a complex number is equal to its magnitude, you can simplify the expression further to:

\left|(\alpha + k)^{2}\right|^{2} \cdot 1 + \left|(\alpha - k)^{2}\right|^{2} \cdot 1 = (\alpha + k)^{4} + (\alpha - k)^{4}

This simplification should make it easier to continue with your solution.