(1+i)^6

3 min read Jun 16, 2024

Exploring the Power of Complex Numbers: (1 + i)^6

The expression (1 + i)^6 might seem intimidating at first glance, but with the right tools and understanding of complex numbers, it becomes a manageable calculation. Let's delve into the process of simplifying this expression.

The Power of De Moivre's Theorem

De Moivre's Theorem provides a powerful tool for dealing with powers of complex numbers. It states that:

[ (cos θ + i sin θ) ]^n = cos(nθ) + i sin(nθ)

where:

n is an integer
θ is an angle in radians

To apply this theorem to our expression, we need to convert (1 + i) into polar form:

Converting to Polar Form

Magnitude: The magnitude of a complex number (a + bi) is given by |z| = √(a² + b²). For (1 + i), the magnitude is √(1² + 1²) = √2.
Angle: The angle θ can be found using the arctangent function: θ = arctan(b/a). In this case, θ = arctan(1/1) = π/4.

Therefore, (1 + i) in polar form is: √2 (cos(π/4) + i sin(π/4)).

Applying De Moivre's Theorem

Now, we can apply De Moivre's Theorem to (1 + i)^6:

[√2 (cos(π/4) + i sin(π/4))]^6 = √2^6 (cos(6π/4) + i sin(6π/4))

Simplifying:

= 8 (cos(3π/2) + i sin(3π/2))

Final Result

Finally, we can convert the result back to rectangular form:

= 8 (0 - i) = -8i

Therefore, (1 + i)^6 = -8i.

Key Takeaways

De Moivre's Theorem simplifies the process of raising complex numbers to powers.
Converting complex numbers to polar form is a crucial step before applying De Moivre's Theorem.
Understanding the relationship between rectangular and polar forms is essential for working with complex numbers.

This example demonstrates the elegance and efficiency of using complex numbers and their properties to solve seemingly complicated expressions.