How do you find the absolute value of the complex number 8 + 12i using the complex conjugate?

To find the absolute value of a complex number, we can use the formula:

• If z is a complex number of the form z = a + bi, where a is the real part and b is the imaginary part, then the absolute value (or modulus) of z, denoted as |z|, is given by:

|z| = sqrt(a² + b²)

In this instance, our complex number is 8 + 12i. Here, a = 8 and b = 12.

We will calculate the absolute value as follows:

1. First, square both the real and imaginary parts:
• a² = 8² = 64
• b² = 12² = 144
2. Next, add the results together:
• a² + b² = 64 + 144 = 208
3. Then, take the square root of that sum:
• |z| = sqrt(208)
4. Finally, simplify the square root if possible:
• |z| = sqrt(16 * 13) = 4sqrt(13)

Therefore, the absolute value of the complex number 8 + 12i is |z| = 4√13 or approximately 14.42.