Problem 85 Simplify. $$ i^{32} $$... [FREE SOLUTION]

Chapter 10: Problem 85

Simplify. $$ i^{32} $$

Short Answer

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Step by step solution

- Understand the Properties of i

The imaginary unit $i$ has specific powers that repeat in a cycle. These powers are: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. This cycle repeats every four powers.

- Find the Remainder

Divide the exponent 32 by the cycle length 4 and find the remainder. \[ 32 \div 4 = 8 \] with a remainder of 0. This means $i^{32}$ falls exactly at the end of a cycle.

- Use the Cycle to Simplify

Since the remainder is 0, it aligns with $i^4$. From the properties of $i$, we know $i^4 = 1$. Therefore, $i^{32} = 1$.

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Key Concepts

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

imaginary unit

The imaginary unit, denoted as $i$, is a fundamental concept in complex numbers. It is defined by the equation $i^2 = -1$. This definition means that $i$ is a number whose square is negative one. In mathematics, this can be quite useful, especially when dealing with roots of negative numbers.
Here are the first few powers of $i$:
* $i^1 = i$
* $i^2 = -1$
* $i^3 = -i$
* $i^4 = 1$
Notice that after $i^4 = 1$, the powers of $i$ start repeating in the same sequence. This repetition forms the basis for understanding cyclic patterns in the powers of the imaginary unit.

cyclic patterns

A cyclic pattern is a sequence of values that repeat at regular intervals. For the imaginary unit $i$, the cycle length is 4, meaning every four powers, the pattern repeats.
This cyclical behavior can be observed as:
* $i^1 = i$
* $i^2 = -1$
* $i^3 = -i$
* $i^4 = 1$
* $i^5 = i$
* $i^6 = -1$
* and so forth...
To simplify large exponents of $i$, you can use this cyclic pattern. For example, to simplify $i^{32}$, recognize that the exponent 32 is a multiple of the cycle length 4. Since $32 \div 4 = 8$ with a remainder of 0, $i^{32}$ falls exactly on $i^4$, which is 1.

remainder theorem for exponents

The remainder theorem for exponents can help simplify powers of $i$. When you divide an exponent by the cycle length, the remainder tells you where you are within the cycle. Here鈥檚 how it works:
* Find the cycle length. For $i$, the length is 4.
* Divide the exponent by this length and find the remainder.
For $i^{32}$:
\[ 32 \div 4 = 8 \quad \text{remainder} = 0 \]
This means $i^{32}$ corresponds to $i^0$ (since the remainder is 0), which simplifies to 1.
Using the remainder theorem, you can quickly identify which power of $i$ corresponds to any large exponent, making it easier to work with complex numbers.

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91影视

Simplify. $$ i^{32} $$

Short Answer

Step by step solution

- Understand the Properties of i

- Find the Remainder

- Use the Cycle to Simplify

Key Concepts

imaginary unit

cyclic patterns

remainder theorem for exponents

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