Question

Find one or more values of each of the following complex expressions in the easiest way you can. \(\left(\frac{1+i}{1-i}\right)^{2718}\)

Short answer

-1

Step-by-step solution

Simplify the Fraction

First, simplify the fraction \(\frac{1+i}{1-i}\) by multiplying the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \(1-i\) is \(1+i\). So we get: \[ \frac{(1+i)(1+i)}{(1-i)(1+i)} = \frac{1+2i+i^2}{1-i^2}. \]

Simplify the Expression

Simplify \(1 + 2i + i^2\) and \(1 - i^2\) using the fact that \(i^2 = -1\). \(\therefore 1 + 2i + (-1) = 2i\). The denominator becomes: \(\therefore 1 - (-1) = 1 + 1 = 2\). So the fraction is \[ \frac{2i}{2} = i. \]

Raise to the Power

Now we need to compute \(i^{2718}\). Use the properties of powers of \(i\), noting that \(i^4 = 1\). Therefore, reduce the exponent modulo 4: \[ 2718 \mod 4 = 2. \] Thus, \(i^{2718} = i^2\).

Calculate the Final Result

Since \(i^2 = -1\), the final result is \(-1\).

Key concepts

Complex Conjugate

When dealing with complex fractions, one helpful technique is to use the complex conjugate. The complex conjugate of a complex number is simply the same number but with the sign of the imaginary part changed. For example, the complex conjugate of the number \(1 - i\) is \(1 + i\).

Using the complex conjugate can help us simplify the fraction. By multiplying the numerator and denominator by the complex conjugate of the denominator, we can eliminate the imaginary part from the denominator, making the expression easier to work with.

Here’s how it works in practice: for the fraction \frac{1+i}{1-i}\, multiplying both parts by the conjugate of the denominator, which is \frac{1+i}{1+i}\, yields: \frac{(1+i)(1+i)}{(1-i)(1+i)}\.. This step ensures that the denominator becomes a real number, which simplifies our calculations.

Powers of i

The imaginary unit \(i\) has a cyclical pattern when raised to successive powers. Here are the first few powers of \(i\):

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

These powers repeat every four terms. This means \(i^5 = i\) , \(i^6 = -1\), and so on. When dealing with large exponents like \(i^{2718}\), it is efficient to reduce the exponent modulo 4. For instance, since \(2718 \text{ mod } 4 = 2\), it follows that \(i^{2718} = i^2 = -1\).

Understanding this cycle is crucial for simplifying expressions involving powers of \(i\).

Simplifying Complex Fractions

Simplifying complex fractions involves a few steps, especially when they include imaginary parts. Let’s briefly recap the process:

1. **Multiply by the Complex Conjugate:** Begin by multiplying the numerator and denominator by the complex conjugate of the denominator. For example, to simplify \frac{1+i}{1-i}\.

2. **Simplify the Numerator and Denominator:** Use the fact that \(i^2 = -1\) to simplify the terms. After multiplying, our fraction becomes \frac{(1+i)(1+i)}{(1-i)(1+i)} = \frac{1+2i+i^2}{1-i^2}\ =\frac{2i}{2}\

3. **Reduce:** Finally, simplify the resulting fraction, which in this example gives \(i\).

These steps will help in breaking down and solving complex fraction problems more efficiently.