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Chapter 3: Problem 85

Simplify. $$(-i)^{71}$$

Short Answer

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Step by step solution

01

- Understand the Properties of i

The imaginary unit, represented as i, has cyclical properties when raised to powers. Specifically, the powers of i follow this cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats.
02

- Determine the Cycle for -i

The powers of \(-i\) can also be determined based on the properties of i. The cycle is: \((-i)^1 = -i\), \((-i)^2 = (-i)(-i) = i^2 = -1\), \((-i)^3 = (-i)(-1) = -i^3 = i\), \((-i)^4 = (-i)(i) = i^4 = 1\), and then it repeats.
03

- Find the Remainder of 71 Divided by 4

Since the powers of \(-i\) repeat every 4, find the remainder when 71 is divided by 4: \(71 \div 4 = 17 \) remainder \(3\). Therefore, \((-i)^{71}\) is in the same position in the cycle as \((-i)^3\).
04

- Use the Cycle to Simplify

From the cycle of \(-i\): \((-i)^1 = -i\), \((-i)^2 = -1\), \((-i)^3 = i\), \((-i)^4 = 1\)We see that \((-i)^3 = i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Unit
In the world of complex numbers, the imaginary unit is symbolized by i. The imaginary unit is defined as \(i = \sqrt{-1}\).
This value allows us to work with the square roots of negative numbers, which is not possible with real numbers alone.
For instance, \(i^2 = -1\). This forms the basis of complex numbers, where complex numbers are expressed as \(a + bi\), with a and b being real numbers. The invention of the imaginary unit i allows us to extend the number system and solve equations that would otherwise be unsolvable.
Powers of i
When we talk about the powers of i, we're observing a pattern that repeats every four terms.
Here's the cycle:
  • \(i^1 = i \)
  • \(i^2 = -1 \)
  • \(i^3 = -i \)
  • \(i^4 = 1 \)

After \(i^4 \), the powers of i cycle back to \(i^1 \). Understanding this cyclical behavior is crucial since any higher power of i can be simplified by finding its position within this repeating cycle.
Cyclical Properties
The cyclical properties of i extend to its negative as well. When raised to powers, \(-i \) also follows a cycle:
  • \((-i)^1 = -i \)
  • \((-i)^2 = -1 \)
  • \((-i)^3 = i \)
  • \((-i)^4 = 1 \)

To simplify any power of \(-i \), find the remainder when the exponent is divided by 4. This remainder tells you where the power lies in the cycle.
For example, simplifying \((-i)^{71} \) involves dividing 71 by 4, giving 17 with a remainder of 3. Thus, \((-i)^{71} = (-i)^3 = i \). This cyclical pattern makes working with powers of i and -i straightforward once understood.

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