Problem 83 Find each power of $i$ $$ ... [FREE SOLUTION]

Chapter 10: Problem 83

Find each power of $i$ $$ (-3 i)^{5} $$

Short Answer

Expert verified

The result of $(-3i)^5$ is $-243i$.

Step by step solution

Understand the powers of i

Recall that the imaginary unit $ i $ satisfies $ i^2 = -1 $. From this, we derive that $ i^3 = i^2 \cdot i = -i $ and $ i^4 = i^3 \cdot i = 1 $. Hence, the powers of $ i $ cycle every four: $ i, -1, -i, 1 $.

Separate coefficients and imaginary parts

The expression $(-3i)^5$ can be separated as $(-3)^5 \cdot i^5$. Thus, we focus first on calculating the power of the real coefficient $(-3)$ and separately handle the power of $i$.

Calculate the power of the real coefficient

Calculate $(-3)^5$ as follows: \[ (-3)^5 = (-3) \times (-3) \times (-3) \times (-3) \times (-3) = -243 \].

Calculate the power of i

Using the cycle of $ i$, to find $ i^5 $, notice that $5 \mod 4 = 1$, so $ i^5 = i $.

Combine results

Combine the results of steps 3 and 4: \[ (-3)^5 \cdot i^5 = -243 \cdot i = -243i \].

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Key Concepts

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

Imaginary Unit

In the world of complex numbers, the imaginary unit is a pivotal concept. Denoted by the symbol $ i $, it represents the square root of $-1$. Unlike real numbers, which have positive or zero squares, the imaginary unit is a novel way to handle the square roots of negative numbers. Complex numbers combine real and imaginary parts in the form $ a + bi $, where $ a $ is the real part and $ b $ is the imaginary part. The introduction of $ i $ allows mathematicians and scientists to extend the real number system and solve equations that would otherwise have no solutions in real numbers alone.

$ i^2 = -1 $: the defining property of the imaginary unit.
Imaginary numbers can be used in various fields such as engineering, physics, and mathematics.

Powers of i

Understanding the powers of the imaginary unit $ i $ is crucial for working with complex numbers. Since $ i^2 = -1 $, higher powers of $ i $ can be simplified by recognizing a cyclic pattern:

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

This cycle repeats every four powers, making calculations easier by reducing more complex exponents to one of these four fundamental values. For example, any power of $ i $ can be simplified by taking the exponent mod 4. In the example $ i^5 $, since $ 5 \mod 4 = 1 $, we know $ i^5 = i $. Recognizing and using this cycle can greatly simplify calculations and provide deep insights when handling complex expressions.

Exponentiation

Exponentiation in the context of complex numbers can initially seem daunting, but it becomes approachable by breaking down the process. Consider an expression with complex numbers like $(-3i)^5$. Here's how to manage it:

Separate effects: split the expression into its real and imaginary components $(-3)^5$ and $i^5$.
Calculate the real number power first: $(-3)^5 = -243$.
Handle the imaginary component using the cycle of powers of $ i $: $ i^5 = i $ since $ 5 \mod 4 = 1 $.

After performing these calculations independently, combine the results for the complete expression: \[ (-3i)^5 = (-243) \times i = -243i \]. By organizing complex number exponentiation into manageable parts, the process becomes clear and applicable.

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91影视

Find each power of \(i\) $$ (-3 i)^{5} $$

Short Answer

Step by step solution

Understand the powers of i

Separate coefficients and imaginary parts

Calculate the power of the real coefficient

Calculate the power of i

Combine results

Key Concepts

Imaginary Unit

Powers of i

Exponentiation

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