Problem 92 - Algebra A Combined Function

Question

Write each expression in the form \(a+b i .\) $$ i^{4}+i^{12} $$

Step-by-Step Solution

Answer

The expression simplifies to \(2 + 0i\).

Step 1: Recognize Patterns in Powers of i

The powers of the imaginary unit, \( i \), cycle every four: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \). Recognize this pattern as key to simplifying powers of \( i \).

Step 2: Simplify \( i^4 \)

Since \( i^4 = 1 \), we replace \( i^4 \) with \( 1 \).

Step 3: Simplify \( i^{12} \)

Using the cycle of powers of \( i \), divide the exponent (12) by 4 to get the remainder. This gives \( 12 \div 4 = 3 \, (remainder \, 0) \), hence \( i^{12} = i^{0} = 1 \).

Step 4: Combine and Simplify

Combine the simplified terms: \( i^4 + i^{12} = 1 + 1 = 2 \).

Step 5: Write in the Form \( a + bi \)

Since the result is a real number, express it as \( 2 + 0i \) to fit the \( a + bi \) form.

Key Concepts

Powers of Imaginary Unit

The imaginary unit, represented by the letter \(i\), forms the foundation of many complex number calculations. Understanding the powers of \(i\) is essential because they exhibit a cyclical behavior every four steps. Here is the cycle:

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

The pattern repeats after every four exponents. So, any power of \(i\) can be simplified by determining its position within this cycle. To do this, divide the exponent by 4 and consider the remainder:

• Remainder 0 corresponds to \(i^4 = 1\)
• Remainder 1 corresponds to \(i^1 = i\)
• Remainder 2 corresponds to \(i^2 = -1\)
• Remainder 3 corresponds to \(i^3 = -i\)

By applying this rule, expressions like \(i^{12}\) can be easily simplified to \(i^0 = 1\), as the remainder when dividing 12 by 4 is 0.

Simplifying Complex Expressions

Complex expressions often contain powers of \(i\), alongside other numerical values. To simplify, you must first address any powers of \(i\) by applying their cyclical nature. For example:

• In the expression \(i^{4} + i^{12}\), first identify each power of \(i\). Here, both \(i^4\) and \(i^{12}\) simplify to 1.

Once the powers of \(i\) are simplified, look for routine algebraic operations such as addition or subtraction among the remaining numbers. Combine like terms, ensuring every constituent is expressed in its simplest form. This step-by-step approach reduces the complexity of expressions, making it easier to see the underlying real and imaginary parts.

Algebraic Expressions

Algebraic expressions involve combinations of numbers, letters, and operational symbols. They form the basis of simplifying complex expressions, especially within the realm of complex numbers. When working with expressions involving \(i\), the goal is to reduce them to their simplest \(a + bi\) form. Here’s how:

• Identify and simplify powers of \(i\) using their cyclical pattern.
• Apply basic algebraic principles to combine like terms.
• Remember to express your final answer in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part (often \(i\) terms).

This transformation converts what might initially appear complex into a strictly ordered arithmetic expression, facilitating understanding and further calculations.

Real and Imaginary Components

Complex numbers are expressed in the form \(a + bi\), where \(a\) is the real component and \(bi\) represents the imaginary component. In the expression \(i^4 + i^{12}\) simplified to \(2 + 0i\), we have:

• \(a = 2\)
• \(b = 0\)

Here, there is no imaginary part. Understanding how to identify these components is crucial. It allows you to visualize complex numbers geometrically or plot them on the complex plane, where the x-axis represents the real part and the y-axis conveys the imaginary part. Simplification leads us to express these components separately, granting clarity and precision in complex arithmetic and geometry.