Problem 93 - Precalculus Student Solutions Manual 5th

Question

Simplify each power of i. $$i^{-13}$$

Step-by-Step Solution

Answer

i^{-13} = -i

Step 1 - Understanding Powers of i

Recall that the powers of the imaginary unit i repeat in a cycle: i^1 = i i^2 = -1 i^3 = -i i^4 = 1 i^5 = i and so on. Therefore, every fourth power of i returns to 1. This cycle can be used to simplify any power of i.

Step 2 - Express the Given Power in Terms of a Multiple of 4

Rewrite the given exponent -13 as a multiple of 4 plus a remainder. To do this, use the division -13 ÷ 4, which gives a quotient of -4 and a remainder of 3. Thus, i^-13 can be written as: i^{-13} = i^{-16 + 3}

Step 3 - Simplify using the Cycle

Since i^{-16} is a multiple of 4 (i.e., 4*(-4)), i^{-16} = 1. Therefore: i^{-13} = i^{-16+3} = i^3 From the known cycle of i, we have i^3 = -i.

Key Concepts

imaginary unit

The imaginary unit 'i' is a fundamental concept in complex numbers. It is defined as the square root of -1 and is written as:

\[ i = \sqrt{-1} \]

This concept allows us to extend the real number system and solve equations that involve negative square roots. The powers of 'i' exhibit a repeating cycle, which is crucial for simplifying expressions like the one in the exercise. Here are the first few powers of 'i':

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

This pattern shows that the powers of 'i' repeat every four terms.

modular arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' upon reaching a certain value—the modulus. This concept is useful when working with cyclic patterns, like the powers of 'i'.

To find the power of 'i' given an exponent, we can use modular arithmetic to reduce the exponent. For instance, in the example of \(i^{-13} \), we want to reduce \(-13\) to a number between 0 and 3 (since the cycle length is 4):

• Divide -13 by 4 to get the quotient and the remainder.
  -13 ÷ 4 = -4 R3
  
• The remainder (3) tells us which power of 'i' corresponds to \(i^{-13} \). So, \(i^{-13} \) can be simplified using \(i^3 \).

This simplification using modular arithmetic makes it easier to handle powers of 'i'.

cyclic patterns

Cyclic patterns are repeated sequences that occur in regular intervals. Understanding cyclic patterns helps simplify complex expressions.

The powers of 'i' display a cyclic pattern with a cycle length of four. Hence, after every four terms, the cycle repeats. This is evident from the powers of 'i':

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

So, for any integer exponent, you can determine the simplified power of 'i' by finding its position within this cycle. As seen in the solution:

• \(i^{-13} \) is cyclically reduced to \(i^3 \)
• \(i^3 = -i \), which simplifies our original expression

Understanding these repeating sequences helps in quickly evaluating and simplifying powers of 'i'.