Solving Complex Exponential Equations Using De Moivre’s Theorem and Euler’s Formula

Complex exponential equations often require a deep understanding of De Moivre’s Theorem and Euler’s Formula. This article explores how to solve the equation (sqrt{8} i^{50} 3^{49} aib) and find the value of (a^2b^2). We will provide a step-by-step solution and highlight the key mathematical concepts involved.

Step-by-Step Solution

To solve the equation (sqrt{8} i^{50} 3^{49} aib) and find (a^2b^2), we can follow these steps:

1. Convert (sqrt{8} i) to Polar Form

First, we convert (sqrt{8} i) into its polar form. To do this, we calculate the modulus and argument.

Modulus:

(r sqrt{8} i sqrt{8 cdot 1} 3)

Argument:

(theta arctanleft(frac{1}{sqrt{8}}right) arctanleft(frac{1}{2sqrt{2}}right))

Thus, we can express (sqrt{8} i) in polar form as:

(sqrt{8} i 3left(costheta isinthetaright))

2. Raise to the 50th Power Using De Moivre’s Theorem

Using De Moivre’s Theorem, we can raise this expression to the 50th power:

(sqrt{8} i^{50} 3^{50} left(cos(50theta) isin(50theta)right))

3. Equate to the Given Expression

We know that:

(3^{50} left(cos(50theta) isin(50theta)right) 3^{49} aib)

Dividing both sides by (3^{49}), we get:

(3 left(cos(50theta) isin(50theta)right) aib)

This gives us:

(a 3 cos(50theta)) and (b 3 sin(50theta))

4. Calculate (a^2b^2)

Now we calculate (a^2b^2):

(a^2b^2 3^2 cos^2(50theta) sin^2(50theta))

Using the Pythagorean identity:

(cos^2(x) sin^2(x) 1)

We get:

(a^2b^2 9 cdot 1 9)

Thus, the final answer is:

(boxed{9})

Alternative Solution Using Euler’s Formula

We can solve this using Euler’s formula, which states:

(e^{itheta} cos(theta) isin(theta))

The given equation is:

(sqrt{8} i^{50} 3^{49} aib)

Dividing by 349 on both sides, we get:

(3 cdot e^{i50theta} aib)

Thus:

(3 left(cos(50theta) isin(50theta)right) aib)

From here, we can find:

(a 3 cos(50theta)) and (b 3 sin(50theta))

Then, we need to find (a^2b^2):

(a^2b^2 9 left(cos^2(50theta) sin^2(50theta)right))

Again, using the Pythagorean identity:

(cos^2(50theta) sin^2(50theta) 1)