Solving Complex Expressions Involving i and Square Roots

In this article, we will explore the solution to complex expressions involving the imaginary unit i and square roots. We will use Euler's formula and exponential form to simplify and solve such expressions. These concepts are fundamental in understanding complex numbers and their applications in various fields, including electrical engineering, physics, and computer science.

Complex Numbers and Euler's Formula

Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary unit i is defined as i √(-1). One of the most powerful tools for dealing with complex numbers is Euler's formula, which states that eix cos(x) i sin(x). This formula allows us to represent complex numbers in exponential form, which can greatly simplify calculations involving complex numbers.

Example Problem: Simplifying (sqrt{3}i / (1 - sqrt{3}i))^{4n}

Let's consider the expression (sqrt{3}i / (1 - sqrt{3}i))^{4n}. To solve this, we will first convert the expression into its exponential form.

First, we express (sqrt{3}i) and (1 - sqrt{3}i) in exponential form:

(sqrt{3}i) can be written as 2.e(frac{ipi}{6}) (as (sqrt{3}i 2(cos(frac{pi}{6}) isin(frac{pi}{6})))). (1 - sqrt{3}i) can be written as 2.e(-frac{ipi}{3}) (as (1 - sqrt{3}i 2(cos(-frac{pi}{3}) isin(-frac{pi}{3})))).

Now, we can simplify the expression by dividing the two exponential forms:

(frac{2.e^(frac{ipi}{6})}{2.e^(-frac{ipi}{3})}) e^(frac{pi i}{2})

Next, we raise the simplified form to the power of 4n:

(e^(frac{pi i}{2}))^(4n) e^(2npi i)

Since (e^(2npi i) 1), the simplified expression of (sqrt{3}i / (1 - sqrt{3}i))^(4n) is 1.

Further Simplification

Let's further simplify the expression (sqrt{3}i / (1 - sqrt{3}i)^{4n}) · (sqrt{3}i).

We already know that (sqrt{3}i / (1 - sqrt{3}i))^(4n) 1, so the expression simplifies to:

(1) · (sqrt{3}i sqrt{3}i)

This shows that the expression (sqrt{3}i / (1 - sqrt{3}i))^(4n) · (sqrt{3}i) simplifies to (sqrt{3}i).

Conclusion

In this article, we have shown how to simplify complex expressions involving the imaginary unit i and square roots by using Euler's formula and exponential form. The key steps involved conversion of the expression into exponential form, simplification using properties of exponentials, and the application of the fundamental properties of complex numbers. Understanding these concepts is essential for working with complex numbers in various fields, including advanced mathematics, physics, and engineering.