Problem 81 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

(-3i)^5 = -243i.

Step 1: Identify the Properties of i

The imaginary unit \(i\) is defined such that \(i^2 = -1\). Therefore, powers of \(i\) repeat in a cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This cycle repeats every four powers.

Step 2: Simplify the Base

The expression is \((-3i)^5\), where \(-3i\) can be considered as a single unit. First express it as \((-1)^5 \times 3^5 \times i^5\), which simplifies to \(-243i^5\).

Step 3: Find i^5 Using Cycle

From the cycle of powers of \(i\), \(i^5\) is the same as \(i^{4+1} = i^4 \times i^1 = 1 \times i = i\).

Step 4: Calculate the Final Expression

Substitute the last result into the expression: \(-243i^5 = -243i\). Therefore, \((-3i)^5 = -243i\).

Key Concepts

Imaginary Unit

When diving into the world of complex numbers, one of the first concepts you'll encounter is the imaginary unit, denoted as \(i\). This unit is a fundamental building block in the realm of complex numbers. Its defining property is quite intriguing: \(i^2 = -1\). Hence, the imaginary unit helps us extend the real number system to tackle operations that are otherwise not possible, like taking the square root of negative numbers.

A unique feature of \(i\) is the cycle it creates when raised to different powers. This cycle repeats every four exponents:

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

This pattern continues, so for any integer power \(n\), the power of \(i\) is determined by the remainder of \(n\) when divided by 4. Understanding this cyclic pattern is crucial for simplifying expressions involving powers of \(i\).

Complex Numbers

Complex numbers are an extension of the real numbers. They include a component involving the imaginary unit \(i\). A complex number is generally represented in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

These numbers add a second dimension to the number line, allowing for advanced computations. For instance, operations like addition and multiplication can be uniformly applied with real numbers, but need special attention with the involvement of \(i\).


In the context of our problem,

• We encountered an expression involving complex numbers: \((-3i)^5\).
• By treating it as a single complex number, we could systematically decompose the expression, using properties of powers, to simplify it.

The elegance of complex numbers lies in their ability to elegantly solve problems that deal with both real and imaginary parts.

Algebraic Expressions

Algebraic expressions can become quite complex when they involve powers of \(i\). The key to managing these expressions is a solid understanding of the properties of \(i\) and how it behaves in algebraic operations.

Suppose you have an expression like our original exercise, \((-3i)^5\). This expression combines real and imaginary components, requiring a strategic approach for simplification.

• First, break it down using the properties of exponentiation and the cycle of \(i\).
• This means expressing it as \((-1)^5 \times 3^5 \times i^5\).
• The breakdown helps separate the real part, \(-243\), from the imaginary power, \(i^5\), which reduces to \(i\) using the cycle.

When working with algebraic expressions with complex numbers, always remember to simplify and use properties like distribution and factorization. These tools help in obtaining a clean final result.