Problem 87 - A Graphical Approach to Precalculus with Limits

Question

Simplify each expression to \(i, 1,-i,\) or \(-1\) $$\frac{(-1)^{4}}{i^{-16}}$$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Simplifying the Numerator

The numerator of the expression is \[(-1)^4\].Since raising \(-1\) to any even power results in 1, thus, \[(-1)^4 = 1.\]

Step 2: Simplifying the Denominator

The denominator \(i^{-16}\) can be simplified using the periodicity of powers of \(i\). Recall that the powers of \(i\) cycle through 1, \(i\), \(-1\), and \(-i\) in every four cycles. Since the power\(-16\) is divisible by 4, the result is \(1\).Thus, \(i^{-16} = 1.\)

Step 3: Forming the Complete Fraction and Simplifying

Now, substitute the simplified numerator and denominator into the fraction:\[\frac{1}{1} = 1.\]Thus, the expression simplifies to 1.

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their most basic form while retaining their equivalence. This often means working through operations like addition, subtraction, multiplication, and division, or dealing with exponents. In our exercise, the aim is to break down the complex-looking fraction into a simpler answer.

Let's break it down step by step using the given problem of simplifying \[\frac{(-1)^4}{i^{-16}}.\].

• First, we simplify the numerator \((-1)^4\). Since \(-1\) raised to an even power results in 1, we find that \((-1)^4 = 1\).


• Next, we need to simplify the denominator \(i^{-16}\). The solution involves recognizing patterns (as we'll see in more detail in the next section below).


• Once both the numerator and denominator are simplified, the resulting expression is easy to evaluate:

\[\frac{1}{1} = 1.\]
Simplifying expressions is crucial for solving more complex problems because it boils down equations or expressions into manageable forms so you can easily identify patterns or solutions.

Powers of i

Understanding the powers of \(i\) is essential in simplifying expressions involving complex numbers. The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). The progression of powers of \(i\) follows a predictable cycle of four:

• \(i^1 = i\)


• \(i^2 = -1\)


• \(i^3 = -i\)


• \(i^4 = 1\)

After \(i^4\), the powers of \(i\) repeat this cycle, which helps in simplifying higher powers of \(i\), whether positive or negative.

To simplify \(i^{-16}\), notice that 16 is divisible by 4, indicating its place in the cycle that ends or begins with 1, thus \(i^{-16} = 1\). By understanding this cyclical nature, we can quickly determine any power of \(i\) without computation—just identify its position within these four steps.

Even Powers

An important concept in simplifying expressions, especially those involving negative bases or imaginary units, is recognizing patterns when numbers are raised to even powers.
Any negative number raised to an even power results in a positive number. For example:

• \((-1)^{2} = 1\)


• \((-1)^{4} = 1\)


• \((-2)^{6} = 64\)

These results occur because multiplying negative numbers an even number of times cancels out the negatives, turning the expression positive.

For the given problem, applying the principle to \((-1)^{4}\), confirms it simplifies to 1, since any even exponent returns a positive outcome. Understanding this concept helps simplify expressions rapidly and with assurance, knowing these patterns remain true across all regular computations.