Problem 82 - Algebra A Combined Function

Question

Find each power of \(i\). $$ (-2 i)^{7} $$

Step-by-Step Solution

Answer

The result is 128i.

Step 1: Identify the properties of i

The imaginary unit, denoted as \( i \), is defined such that \( i^2 = -1 \). A key cyclical relation is its powers: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). These repeat every four exponents, given the cyclical nature of \( i \).

Step 2: Separate components of the expression

In the expression \((-2i)^7\), separate the real number from the imaginary unit: \(-2^7\) and \(i^7\). Calculate each part separately.

Step 3: Calculate the real number part

Compute the power of the real number part \(-2^7\). Since -2 raised to an odd power keeps the negative sign, calculate: \(-2^7 = -(2^7) = -128\).

Step 4: Compute the power of i

Use the cyclical pattern of \( i \) to find \( i^7 \). Divide 7 by 4 to find the remainder: 7 mod 4 = 3. Therefore, \( i^7 = i^3 = -i \).

Step 5: Combine real and imaginary parts

Combine the results from Steps 3 and 4: \(-128 \cdot (-i) = 128i\). Therefore, \((-2i)^7 = 128i\).

Key Concepts

Powers of Imaginary Unit

Understanding the powers of the imaginary unit, denoted as \( i \), is pivotal in complex number arithmetic. Initially, it is essential to know that \( i \) is the imaginary unit where \( i^2 = -1 \). This unique property defines \( i \) and forms the basis for calculating its powers.

To compute any power of \( i \), leverage the cyclical nature of its powers. Knowing the first few powers can help with this:

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

This sequence repeats every four powers, simplifying calculations significantly. Grasping these basics helps with managing more complex expressions involving \( i \).

Imaginary Number i

The imaginary number \( i \) is a cornerstone of complex number theory. It arises from the solution of the equation \( x^2 + 1 = 0 \). In simple terms, \( i \) enables the square root of negative numbers.

While real numbers represent values along a single number line, complex numbers, which include imaginary parts, extend this concept to a plane. This is called the complex plane where each number has both a real and an imaginary component. The power of \( i \) arises from its ability to rotate or oscillate through a series of predictable transformations. Understanding these properties unveils a new dimension of mathematics that is pivotal in various applications like engineering and physics.

Cyclical Pattern in Powers

One of the intriguing aspects of the imaginary unit \( i \) is its cyclical pattern in powers. After the fourth power, the cycle \([i, -1, -i, 1]\) repeats. This pattern forms the basis of simplification when dealing with powers of \( i \).

To determine the power of any exponent beyond 4, simply divide the exponent by 4 and find the remainder:

• If the remainder is 0, \( i^n = 1 \)
• If the remainder is 1, \( i^n = i \)
• If the remainder is 2, \( i^n = -1 \)
• If the remainder is 3, \( i^n = -i \)

This cyclical nature simplifies and speeds up calculations remarkably, making it easier to handle complex powers without extensive computation.

Simplifying Complex Powers

Simplifying complex powers involving \( i \) combines understanding both real number arithmetic and the cyclical pattern of \( i \). Consider the expression \((-2i)^7\) as an example.

Break it down into two distinct parts:

• Calculate the real component: \((-2)^7\), which equals \(-128\).
• Determine the power of \( i \) using its cyclical pattern: \(i^7 = i^3 = -i\).

Finally, multiply these parts back together: \(-128 \times (-i) = 128i\).

The steps involve identifying the nature of the powers involved and simplifying using cyclical and arithmetic properties. Mastery of this technique is essential for working with complex numbers efficiently.