Problem 95 Simplify. $$i^{2}+i^{4}$$... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 95

Simplify. $$i^{2}+i^{4}$$

Short Answer

Expert verified

Step by step solution

- Write Down the Values of i^2 and i^4

Recall that in complex numbers, the imaginary unit satisfies the equation ewline ewline$i^{2} = -1$. ewline ewlineTo find the value of $i^4$: ewline - Start by recognizing that \[i^4 = (i^2) \times (i^2) = (-1) \times (-1)\] ewline - This simplifies to $1$.

- Substitute the Known Values

Substitute $i^2 = -1$ and $i^4 = 1$ into the original expression: $i^2 + i^4$ becomes ewline ewline$-1 + 1$.

- Simplify the Expression

Combine the terms: ewline ewline $-1 + 1 = 0$. ewline Therefore, the simplified form of the expression $i^2 + i^4$ is ewline $0$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex numbers

Let's start by understanding what complex numbers are. Complex numbers are numbers that have two parts: a real part and an imaginary part. They are written in the form of $a + bi$, where $a$ and $b$ are real numbers, and $i$ represents the imaginary unit. Complex numbers help us solve equations that have no real solutions.

For example, the square root of a negative number is impossible in the real number system. But in the system of complex numbers, we can define such roots using the imaginary unit.

imaginary unit

The imaginary unit, denoted as $i$, is a foundational concept in complex numbers. By definition, $i$ is the square root of -1.

Here are some of its crucial properties:

$i^2 = -1$
$i^3 = i^2 \times i = -1 \times i = -i$
$i^4 = i^2 \times i^2 = -1 \times -1 = 1$

Notice that after every four powers, the values repeat, which is essential when we simplify expressions involving higher powers of $i$ as we did in the original exercise.

exponentiation

Exponentiation is a mathematical operation involving numbers raised to a power. When dealing with the imaginary unit $i$ and its powers, it's helpful to know the repeating pattern:

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

This cycle repeats for higher powers of $i$. Understanding this cycle helps simplify expressions involving $i$ easily. For example, in the exercise when simplifying $i^2 + i^4$, we replaced $i^2$ with -1 and $i^4$ with 1 to simplify the expression to $0$.

Remember, knowing these properties makes working with complex numbers and the imaginary unit straightforward.

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91影视

Simplify. $$i^{2}+i^{4}$$

Short Answer

Step by step solution

- Write Down the Values of i^2 and i^4

- Substitute the Known Values

- Simplify the Expression

Key Concepts

complex numbers

imaginary unit

exponentiation

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