Problem 91 - Intermediate Algebra

Question

Find each power of \(i .\) See Example 6. $$ (-3 i)^{5} $$

Step-by-Step Solution

Answer

The expression \((-3i)^5 = -243i\).

Step 1: Simplify the expression

First, identify the base of the expression, which is \((-3i)^5\). Notice that the term within parentheses is \(-3i\). To simplify it, it can be separated as \((-1)^5 \cdot 3^5 \cdot i^5\). This step is crucial because it leverages the power rule.

Step 2: Calculate \((-1)^5\)

Since raising \(-1\) to an odd power \(n\) results in \(-1\), compute: \((-1)^5 = -1\).

Step 3: Calculate \(3^5\)

Use the property of exponents to calculate powers of numbers: \(3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243\).

Step 4: Determine \(i^5\)

First, recall that \(i^2 = -1\). From this calculation, we find \(i^4 = (i^2)^2 = -1 \times -1 = 1\). Thus, \(i^5 = i^4 \times i = 1 \times i = i\). This means for powers of \(i\) that are greater than \(4\), you can reduce the power modulo \(4\) to find the equivalent power less than \(5\), where the cycle of powers starts over.

Step 5: Combine all parts to get the solution

Combine the results from the previous calculations: \((-1) \times 243 \times i = -243i\). Thus the simplified expression for \((-3i)^5\) is \(-243i\).

Key Concepts

Powers of i

Imaginary numbers revolve around the unit imaginary number, denoted as \(i\), which is defined by the equation \(i^2 = -1\). Understanding the powers of \(i\) is fundamental when dealing with complex numbers and their computations. The powers of \(i\) follow a predictable cyclical pattern because, beyond \(i^2\), they start to repeat every four exponents:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• Then the cycle repeats: \(i^5 = i\), \(i^6 = -1\), and so forth.

Recognizing this cycle helps in simplifying expressions involving powers of \(i\). If you encounter a higher exponent, such as \(i^5\), you can reduce it by dividing the exponent by 4 and taking the remainder, utilizing the repeating cycle for easy computation.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They take the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part, while \(bi\) is the imaginary part.
The use of complex numbers is vital in mathematics because they enable us to handle equations that cannot be solved using only real numbers. For example, the equation \(x^2 + 1 = 0\) has no real solutions, but it has solutions in the set of complex numbers: \(x = i\) and \(x = -i\).
To operate with complex numbers, you can:

• Add by combining like terms (real with real, imaginary with imaginary).
• Multiply using the distributive property and simplify using \(i^2 = -1\).

Such operations make complex numbers a powerful tool in both theoretical and applied mathematics.

Exponents and Powers

Exponents and powers form the backbone of mathematical operations involving repeated multiplication. If you have a number \(a\) raised to the power of \(n\), denoted as \(a^n\), the expression means multiplying \(a\) by itself \(n\) times. This is especially useful for compactly representing large calculations.
Powers have several fundamental properties that are frequently used:

• \(a^m \cdot a^n = a^{m+n}\) — When multiplying like bases, add the exponents.
• \((a^m)^n = a^{m \cdot n}\) — When raising a power to another power, multiply the exponents.
• \(a^0 = 1\) (assuming \(aeq 0\)) — Any nonzero number raised to the power of zero is one.

These attributes make working with exponents efficient and form the basis for simplifying complex mathematical expressions. Understanding these principles is crucial when solving equations, like breaking down powers of \(-3i\) in multifaceted expressions.