Question

Perform the indicated operation, and write each expression in the standard form \(a+\) bi. $$ i^{8}+i^{6}-i^{4}-i^{2} $$

Short answer

0 + 0i

Step-by-step solution

Simplify each power of i

The first step is to simplify each power of the imaginary unit, i. Recall that the imaginary unit, i, has a cyclical pattern: i^1 = ii^2 = -1i^3 = -ii^4 = 1. Using this pattern, simplify the higher powers:i^8 = (i^4)^2 = 1^2 = 1,i^6 = (i^4)(i^2) = (1)(-1) = -1,i^4 = 1, i^2 = -1.

Substitute the values back into the expression

Replace each power of i with its simplified value in the given expression:i^{8} + i^{6} - i^{4} - i^{2}Substituting the values from Step 1:1 + (-1) - 1 - (-1)

Perform the arithmetic operations

Perform the arithmetic operations as follows:1 + (-1) - 1 - (-1)Simplify each step:1 - 1 = 0 0 - 1 = -1-1 + 1 = 0

Write the final answer in standard form

The final value is 0, which in standard form is written as:0 + 0i

Key concepts

imaginary unit

The concept of the imaginary unit is fundamental to understanding complex numbers. The imaginary unit, denoted as i, is defined by the property that:

i^2 = -1

This definition allows us to extend the real number system to include complex numbers.

The imaginary unit does not have a real value but helps in solving equations that wouldn’t have solutions otherwise, such as \( ... x^2 + 1 = 0 ... \).

The introduction of i leads to the creation of complex numbers, which are expressed in the form \( ... a + bi ... \), where

a and b are real numbers, and b is the coefficient of the imaginary unit, i.

The expression consists of a real part (a) and an imaginary part (bi).

For example, in the complex number \( ... 3 + 4i ... \), 3 is the real part, and 4i is the imaginary part.

powers of i

When working with complex numbers, understanding the powers of the imaginary unit i is crucial. The powers of i follow a specific cyclical pattern that repeats every four powers:

• \( ... i^1 = i ... \)
• \( ... i^2 = -1 ... \)
• \( ... i^3 = -i ... \)
• \( ... i^4 = 1 ... \)

This pattern continues indefinitely. For example:

• \( ... i^5 = i ... \)
• \( ... i^6 = -1 ... \)
• \( ... i^7 = -i ... \)
• \( ... i^8 = 1 ... \)

Understanding this cycle helps simplify complex expressions involving powers of i. For example, in the given exercise, powers like \( ... i^8 ... \) and \( ... i^6 ... \) can be simplified using their cyclical values:

• \( ... i^8 = 1 ... \)
• \( ... i^6 = -1 ... \)

standard form

Writing complex numbers in standard form makes it easier to work with and understand them. The standard form for a complex number is expressed as:

\( ... a + bi ... \)

• \( ...a ... \) is the real part of the complex number
• \( ...bi ... \) is the imaginary part of the complex number

In the given exercise, after simplifying the powers of i and performing the arithmetic operations, the result was 0. In standard form, this is written as:

\( ... 0 + 0i ... \)

Standard form helps in basic operations like addition, subtraction, multiplication, and division of complex numbers. Additionally, it clarifies that there is both a real part (0) and an imaginary part (0i), even when one or both of these parts are zero.