Problem 42 - College Algebra

Question

For the following exercises, use a calculator to help answer the questions. Evaluate \((1+i)^{k}\) for \(k=4,8,\) and \(12 .\) Predict the value if \(k=16\).

Step-by-Step Solution

Answer

For \(k=4, 8, 12, 16\), the values are \(-4, 16, -64, 256\).

Step 1: Understanding the Expression

We need to evaluate \((1+i)^k\) where \(k\) is 4, 8, and 12. Here, \(i\) represents the imaginary unit, i.e., \(i^2 = -1\).

Step 2: Calculation for k=4

First, calculate \((1+i)^4\). We know that \((1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i\), therefore, \((1+i)^4 = (2i)^2 = 4i^2 = 4(-1) = -4\).

Step 3: Calculation for k=8

Next, calculate \((1+i)^8\). Since \((1+i)^4 = -4\), it follows that \((1+i)^8 = ((1+i)^4)^2 = (-4)^2 = 16\).

Step 4: Calculation for k=12

Now calculate \((1+i)^{12}\). Since \((1+i)^4 = -4\) and \((1+i)^8 = 16\), we have \((1+i)^{12} = (1+i)^8 \times (1+i)^4 = 16 \times (-4) = -64\).

Step 5: Predicting the value for k=16

Finally, predict the value for \((1+i)^{16}\). Since \((1+i)^4 = -4\), it follows that \((1+i)^{16} = ((1+i)^4)^4 = (-4)^4 = 256\).

Key Concepts

Imaginary Unit

In complex numbers, the imaginary unit, denoted by \(i\), is a fundamental concept. It is used to express numbers that go beyond the real number line. The imaginary unit is defined by the property \(i^2 = -1\). This means when you multiply \(i\) by itself, you get -1. Consequently, complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers.

• \(a\) is the real part.
• \(bi\) is the imaginary part.

Understanding the imaginary unit is crucial for calculations involving complex numbers, as it allows for operations that extend beyond the real numbers, handling roots of negative numbers with ease.

Exponentiation of Complex Numbers

Exponentiation is the process of raising a number to a power. When this involves complex numbers, especially those containing the imaginary unit \(i\), it adds some exciting layers. Take, for example, a complex number like \(1+i\). To find powers of this number like \((1+i)^k\), fundamental arithmetic can be used alongside properties of \(i\).
By expanding, substituting \(i^2 = -1\), and simplifying, one can compute powers accurately. For instance, \((1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i\). This process is systematic and requires attention to the roles played by both the real and imaginary components.

Pattern Prediction in Complex Powers

When you calculate higher powers of complex numbers, patterns often emerge. Recognizing these patterns can simplify computations significantly. In the given example, notice how \((1+i)\) is raised to successively higher powers, forming a pattern:

• \((1+i)^4 = -4\)
• \((1+i)^8 = 16\)
• \((1+i)^{12} = -64\)

Next, predicting the value of \((1+i)^{16}\) becomes easier; the steps used in smaller powers can be applied repeatedly. Recognizing that \((1+i)^4 = -4\), you then calculate \((-4)^4 = 256\). This predictability saves time and effort in evaluating complex powers beyond simple multiplication.