Problem 104 - Intermediate Algebra

Question

Write in the form \(a+b i\). $$ i^{8}-i^{7} $$

Step-by-Step Solution

Answer

The expression simplifies to \(1 + i\).

Step 1: Calculate Powers of i

Recall the powers of the imaginary unit \(i\): - \(i^1 = i\)- \(i^2 = -1\)- \(i^3 = -i\)- \(i^4 = 1\), and the cycle repeats every four powers.Using this cycle, calculate \(i^8\) and \(i^7\).

Step 2: Calculate i^8

Since the powers of \(i\) repeat every four terms, \(i^8\) is equivalent to \((i^4)^2 = 1^2 = 1\).

Step 3: Calculate i^7

Similarly, \(i^7 = i^{4+3} = i^4 \cdot i^3 = 1 \cdot (-i) = -i\).

Step 4: Subtract the Imaginary Numbers

Subtract \(i^7\) from \(i^8\) to get the expression in the form \(a + bi\):\(i^8 - i^7 = 1 - (-i) = 1 + i\).

Step 5: Present the Result

The expression \(i^8 - i^7\) simplifies to \(1 + i\), which is in the form \(a+bi\) with \(a=1\) and \(b=1\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It's defined as the square root of \(-1\). This might seem unusual because there is no real number whose square gives \(-1\). That's where \(i\) comes in handy, providing a way to work with squares of negative numbers.
The essential property of the imaginary unit is:

• \(i^2 = -1\)

Understanding \(i\) is crucial as it allows us to expand the number system from real numbers to complex numbers, making it possible to solve equations that have no real solutions.

Powers of i

To master complex numbers, you need to understand how the powers of \(i\) cycle every four terms. Memorizing the pattern can simplify numerous calculations.

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

Then it repeats: \(i^5 = i\), \(i^6 = -1\), and so on. Knowing this cycle allows you to effortlessly find higher powers, like in the given problem where \(i^8\) and \(i^7\) are evaluated. Recognizing these cycles helps reduce complex expressions involving powers of \(i\).

Expressions in a+bi Form

The expression \(a+bi\) is the standard form for complex numbers. Here, \(a\) and \(b\) are real numbers, where \(a\) is the real part, and \(bi\) is the imaginary part of the complex number. This form enables easy arithmetic and visualization on the complex plane.
Given the expression \(i^8 - i^7\), it simplifies to \(1 + i\) in the \(a+bi\) form. By calculating the powers first and subtracting, the expression converts seamlessly into the standard complex number format. Remember, always express your final answer in \(a+bi\) when dealing with complex numbers to make the results clear and structured.