Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Simplify each expression to \(i, 1,-i,\) or \(-1\) $$i^{15}$$

Step-by-Step Solution

Answer

\(i^{15} = -i\).

Step 1: Understand Powers of i

The imaginary unit \(i\) has a property where its powers repeat in a set cycle. Specifically, \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). After the fourth power, these values repeat in a cycle: \(i^5 = i\), \(i^6 = -1\), and so on.

Step 2: Determine Cycle Position

To find \(i^{15}\), we need to determine its position in the repeating cycle by finding the remainder when 15 is divided by 4 (the length of the cycle). Compute \(15 \div 4\) which equals 3 with a remainder of 3. Therefore, \(i^{15}\) has the same value as \(i^3\).

Step 3: Use the Cycle to Simplify

Recall from Step 1 that \(i^3 = -i\). Since \(i^{15}\) corresponds to \(i^3\) in the cycle, we conclude that \(i^{15} = -i\).

Key Concepts

Powers of i

When dealing with imaginary numbers, one key component is understanding the powers of the imaginary unit, denoted as \( i \). The number \( i \) is defined as the square root of -1, which may seem strange as no real number squared gives a negative result. This is where imaginary numbers come into play.

The powers of \( i \) form a cycle that repeats every four terms. This is because:

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

This repeating pattern makes calculating higher powers simple. For example, \(i^5\) is equivalent to \(i\) because it extends beyond \(i^4\) by just one position in the cycle.

Every time the exponent increases by four, the cycle starts over. It is like the hands of a clock returning to the 12 o'clock position after completing a full rotation.

Complex Numbers

Complex numbers are an extension of the real numbers and include the imaginary unit \(i\). A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

• The term \(a\) is referred to as the real part.
• The term \(bi\) is known as the imaginary part since it contains \(i\).

Complex numbers are used extensively in engineering, physics, and applied mathematics to solve problems that involve square roots of negative numbers.

They allow calculations in a higher plane by combining both real and imaginary parts. With complex numbers, calculations such as addition, subtraction, and multiplication also require combining the like terms, ensuring the real and imaginary components are dealt with properly.

Simplifying Expressions

Simplifying expressions involving powers of \(i\) becomes easier when you understand the repeating cycle. Let's use the example of \(i^{15}\).

To simplify this expression, you first need to check the power against the cycle length, which is four. You divide the exponent, 15 in this case, by 4. This gives you a quotient of 3 with a remainder of 3.

This remainder indicates the equivalent power of \(i\) within the cycle. So, \(i^{15}\) corresponds to \(i^3\) in the cycle, which is \(-i\).

• Identify the cycle position using division by 4.
• Use the remainder to determine the power within the cycle.
• Substitute with the corresponding value from the cycle.

This method simplifies evaluating expressions with large exponents of \(i\), allowing quicker and more efficient calculations without individually multiplying \(i\) 15 times.