Problem 83 - A Graphical Approach to Precalculus with Limits

Question

Simplify each expression to \(i, 1,-i,\) or \(-1\) $$\frac{1}{i^{-51}}$$

Step-by-Step Solution

Answer

\(-i\)

Step 1: Simplify the Negative Exponent

To simplify \(\frac{1}{i^{-51}}\), we first note that raising \(i\) to a negative exponent results in the reciprocal of \(i\) raised to the positive of that exponent. Thus, \(i^{-51} = \frac{1}{i^{51}}\). Therefore, the original expression becomes \(1 \div \frac{1}{i^{51}} = i^{51}\).

Step 2: Determine the Pattern of i's Powers

The powers of \(i\) follow a cyclic pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats. So, every four powers, the pattern repeats. This means we need to find \(i^{51}\) by seeing where 51 falls in this cycle.

Step 3: Find Equivalent Power in Cycle

Since the powers of \(i\) cycle every four terms, find the remainder of 51 when divided by 4. Computing \(51 \div 4\), we find \(51 = 4 \times 12 + 3\). The remainder is 3, which means \(i^{51} = i^3\).

Step 4: Use the Power to Simplify

As determined in Step 2, \(i^3 = -i\). Therefore, \(i^{51} = -i\). Thus, the expression simplifies to \(-i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as \(i = \sqrt{-1}\). This might seem a bit perplexing initially since the square root of a negative number is not a real number.

The imaginary unit helps us extend the real number line into the complex plane, allowing solutions to equations that would otherwise be impossible over real numbers.

• It serves as the building block for imaginary and complex numbers.
• In the complex number system, numbers take the form \(a + bi\), where \(a\) and \(b\) are real numbers.
• The role of \(i\) is to create a perpendicular axis to the real numbers on a graph, giving a two-dimensional space where complex numbers "live".

Understanding \(i\) is crucial as it opens up new ways to solve problems and creates avenues for mathematical innovation beyond standard algebra.

Negative Exponents

Understanding negative exponents is essential in mathematics. They represent the reciprocal of the base raised to the corresponding positive exponent. For example, \(a^{-n} = \frac{1}{a^n}\).

• This rule doesn't change when the base is the imaginary unit \(i\).
• For instance, \(i^{-51} = \frac{1}{i^{51}}\). This shift from a negative to a positive exponent simplifies manipulation of expressions in problem-solving.
• It's also a key step in transitioning from negative powers to finding manageable forms, aiding in simplification processes such as the one in the given exercise.

By comprehending negative exponents, students gain skills that help them tackle more complex algebraic operations, particularly those involving cycles and repetitive patterns like the powers of \(i\).

Cyclic Patterns in Mathematics

Cyclic patterns are common in mathematics, offering efficiency and predictability. The powers of imaginary unit \(i\) demonstrate a clear cyclic pattern that recurs every four stages:

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

Once the fourth power is reached, the cycle repeats itself. Recognizing this cycle allows us to express higher powers, like \(i^{51}\), in simpler terms by only considering the remainder when dividing by 4.

Cyclic patterns are a powerful tool:

• They help simplify expressions efficiently, reducing computational effort.
• Awareness of such patterns allows students to approach problems strategically rather than through rote calculation.
• They are not limited to complex numbers but appear in various mathematical contexts, offering students more general strategic insights.

Becoming familiar with these patterns enhances comprehension and fluency across multiple areas of mathematics.