Problem 106 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

2 + 0i

Step 1: Calculate powers of i

To simplify expressions involving powers of the imaginary unit \(i\), we need to understand that \(i^2 = -1\). Next, observe the pattern: \(i^3 = -i\), \(i^4 = 1\), and \(i^5 = i\), which repeats every four powers. Therefore, \(i^4 = 1\).

Step 2: Calculate \(i^{12}\)

Since the powers of \(i\) repeat every four terms, we can express \(i^{12}\) in terms of \(i^{4}\). Specifically, \(i^{12} = (i^4)^3 = 1^3 = 1\).

Step 3: Add the results

Now that we have \(i^4 = 1\) and \(i^{12} = 1\), we can add these results together: \(1 + 1 = 2\).

Step 4: Write in form \(a + bi\)

The result from the previous step is a real number 2. Since there is no imaginary part, in the form \(a + bi\), it is equivalent to \(2 + 0i\).

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers. A complex number is composed of a real part and an imaginary part and can be written in the form \(a + bi\). Here, \(a\) is the real part, and \(b\) is the coefficient of the imaginary unit \(i\).
The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This definition leads to interesting applications in algebra and engineering, helping solve problems that involve square roots of negative numbers.
Complex numbers can be visualized on the complex plane where the x-axis represents the real part and the y-axis represents the imaginary part. This two-dimensional representation helps in understanding the magnitude and the direction of complex numbers, making concepts like addition and multiplication more intuitive.

Powers of i

Understanding powers of \(i\) is key to simplifying expressions like those involving complex numbers. The imaginary unit has a cyclic nature in its powers due to the defining relationship \(i^2 = -1\).
Here's how the cycle works:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• \(i^5 = i\), and the cycle repeats

This means any power of \(i\) can be reduced to one of these four results. By recognizing this periodic cycle, complex algebraic expressions become easier to manage. For instance, \(i^{12}\) simplifies to \((i^4)^3 = 1^3 = 1\), as it matches the cycle.

Simplification of Expressions

Simplifying expressions involving \(i\) often relies on understanding and applying the cycle of its powers. When faced with high powers of \(i\), break them down using multiples of 4, because \(i^4 = 1\). This greatly reduces the complexity of calculations.
To simplify an expression like \(i^{12} + i^{4}\), recognize \(i^{12}\) can be written as a multiple of \(i^4\):

• \(i^{12} = (i^4)^3 = 1\)
• \(i^4 = 1\)

After simplification, add the terms together as you would with real numbers: \(1 + 1 = 2\). Keeping these simplifications in mind helps standardize complex expressions into a more workable form.

Algebraic Expressions

Algebraic expressions with complex numbers follow many of the same rules as those with real numbers, but with the added consideration of \(i\). When writing the results of these expressions, it's crucial to express them in the standard form \(a + bi\).
For our exercise, simplifying \(i^{4} + i^{12}\) resulted in 2, a real number with no imaginary component. To write this in the form \(a + bi\), you assign \(a\) the value 2 and \(b\) the value 0, giving \(2 + 0i\).
This form maintains the structure of a complex number even when the imaginary part is zero, highlighting the potential for complex number calculations even in seemingly straightforward scenarios.