Problem 88 Simplify. $$ i^{64} $$... [FREE SOLUTION]

Chapter 10: Problem 88

Simplify. $$ i^{64} $$

Short Answer

Expert verified

The simplified form of $i^{64}$ is 1.

Step by step solution

Recall the Powers of i

The imaginary unit i has a cyclic pattern for its powers: \[ i^1 = i, \ i^2 = -1, \ i^3 = -i, \ i^4 = 1 \] This cycle repeats every four powers.

Find the Remainder of the Exponent Divided by 4

To determine where in the cycle the power of i falls, divide the exponent by 4 and find the remainder: \[ 64 \div 4 = 16 \text{ with a remainder of } 0 \] This means that $i^{64}$ falls at the same position in the cycle as $i^0$.

Simplify Using the Cycle

From the cycle: \[ i^0 = 1 \] Therefore, $i^{64} = 1$.

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Key Concepts

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

Imaginary Unit

In algebra, complex numbers are employed to extend the concept of one-dimensional number lines to two dimensions. The imaginary unit, denoted as 'i', is defined by the fundamental equation: \[ i^2 = -1 \]This means that 'i' is the square root of -1. Utilizing 'i', we can represent complex numbers in the form of $ a + bi $, where 'a' and 'b' are real numbers. This notation helps handle mathematical operations involving square roots of negative numbers effectively. Although 'i' might initially seem confusing, mastering it is essential for solving various algebraic problems and understanding higher-level mathematics.

Cyclic Pattern

When dealing with powers of 'i', you will notice a repeating pattern, which simplifies computations significantly. The cyclic nature of 'i' revolves around four distinct powers:

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

This pattern repeats every four exponents. For example, $ i^5 $ is the same as $ i^1 $, $ i^6 $ is the same as $ i^2 $, and so forth. Recognizing this cyclic pattern allows you to determine the value of any power of 'i' quickly. For instance, to find $ i^{64} $, you simply divide 64 by 4, resulting in a remainder of 0. Therefore, $ i^{64} = i^0 = 1 $.

Exponentiation of Complex Numbers

Exponentiation involving complex numbers follows similar rules to real numbers but gets an interesting twist due to the presence of 'i'. When raising 'i' to any integer power, the cyclic pattern greatly simplifies calculations. To exponentiate 'i', observe the power and reduce it modulus 4, since the cyclic pattern repeats every 4 numbers. For example:

$ i^{10} $ reduces to $ i^2 $ because $ 10 \% 4 = 2 $, giving $ i^{10} = -1 $
$ i^{123} $ reduces to $ i^3 $ because $ 123 \% 4 = 3 $, giving $ i^{123} = -i $

This exponent reduction method helps keep complex number calculations simple and manageable. Understanding this principle is fundamental to mastering algebraic operations involving imaginary and complex numbers.

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91影视

Simplify. $$ i^{64} $$

Short Answer

Step by step solution

Recall the Powers of i

Find the Remainder of the Exponent Divided by 4

Simplify Using the Cycle

Key Concepts

Imaginary Unit

Cyclic Pattern

Exponentiation of Complex Numbers

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