# Calculating complex number

• Theofilius
In summary, the conversation discusses how to calculate the complex number i raised to the power of 3/4. The solution is to use the laws of exponents and convert i to its exponential form, resulting in a non-unique answer due to the non-uniqueness of square roots.

#### Theofilius

[SOLVED] calculating complex number

## Homework Statement

Calculate $${i}^{\frac{3}{4}}$$

## The Attempt at a Solution

I tried with

$$i=cos\frac{pi}{2}+isin\frac{\pi}{2}$$

$$i^3=cos\frac{3pi}{2}+isin\frac{3\pi}{2}$$

$$i^3=-i$$

$$\sqrt[4]{i^3}=\sqrt[4]{-i}$$

Why don't you just go all the way at once? i=e^(i*pi/2). So i^(-3/4)=(e^(i*pi/2))^(-3/4). Now use laws of exponents. Notice this answer isn't unique. i is also equal to e^(i*(pi/2+2pi)=e^(i*5pi/2). This is the same sort of nonuniqueness you get with square roots.

Thanks. I solve it.

## What is a complex number?

A complex number is a number that contains both a real part and an imaginary part. It is typically written in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary part is represented by the letter i, which is defined as the square root of -1.

## How do you calculate the absolute value of a complex number?

The absolute value of a complex number is equal to the distance between the number and the origin on the complex plane. It can be calculated by taking the square root of the sum of the squares of the real and imaginary parts. For example, the absolute value of the complex number 3 + 4i would be √(3^2 + 4^2) = √25 = 5.

## What is the conjugate of a complex number?

The conjugate of a complex number is found by changing the sign of the imaginary part. For example, the conjugate of the complex number 3 + 4i would be 3 - 4i. This is important in complex number calculations because multiplying a complex number by its conjugate results in a real number.

## How do you add or subtract complex numbers?

To add or subtract complex numbers, simply add or subtract the real parts and the imaginary parts separately. For example, to add the complex numbers (3 + 4i) and (2 + 5i), you would add 3 + 2 = 5 for the real part and 4 + 5 = 9 for the imaginary part, resulting in the complex number 5 + 9i.

## What is the difference between multiplying and dividing complex numbers?

When multiplying complex numbers, you multiply the real parts and the imaginary parts separately, and then combine them to get the final result. For dividing complex numbers, you need to first multiply the numerator and denominator by the conjugate of the denominator, which will result in a real number in the denominator. Then, you can divide the real and imaginary parts separately to get the final result.