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[abc159835]

[abc159835]

Raw content is: Great question! Let's delve into why (|z|^n = |z^n|) for a complex number (z) and a positive integer (n).

First, recall that for a complex number (z = a + bi), where (a) and (b) are real numbers, the magnitude (or modulus) (|z|) is defined as:

[ |z| = \sqrt{a^2 + b^2} ]

Now, let's consider the complex number (z) raised to the power of (n):

[ z^n = (a + bi)^n ]

The magnitude of (z^n), denoted as (|z^n|), is the modulus of the complex number (z^n).

One of the key properties of the modulus of a complex number is that it is multiplicative. This means that for any two complex numbers (z_1) and (z_2):

[ |z_1 \cdot z_2| = |z_1| \cdot |z_2| ]

Using this property, we can extend it to (n) factors of the same complex number (z):

[ |z^n| = |z \cdot z \cdot \ldots \cdot z| \quad \text{(n times)} ]

Applying the multiplicative property of the modulus repeatedly, we get:

[ |z^n| = |z| \cdot |z| \cdot \ldots \cdot |z| \quad \text{(n times)} ]

This simplifies to:

[ |z^n| = (|z|)^n ]

So, we have shown that:

[ |z^n| = |z|^n ]

This equality holds because the modulus of a product of complex numbers is the product of their moduli, and this property extends naturally to any integer power of a complex number.