Problem 77 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \frac{8}{1+\frac{2}{i}} $$

Step-by-Step Solution

Answer

The result in standard form is \( \frac{8}{5} - \frac{16}{5}i\).

Step 1: Identify the Conjugate

The conjugate of a complex number flips the sign between the two terms. So, the conjugate of \(1 + \frac{2}{i}\) is \(1 - \frac{2}{i}\).

Step 2: Multiply the fraction by the conjugate

Multiply the numerator and denominator by the conjugate \(1 - \frac{2}{i}\). Hence, \(\frac{8}{1+\frac{2}{i}} \times \frac{1 - \frac{2}{i}}{1 - \frac{2}{i}}\).

Step 3: Multiply the numerators and denominators

Multiply across the top and the bottom of the fraction. Hence, \(8 \times (1 - \frac{2}{i})\) gives \(8 - 16i\). And on calculating the denominator: \( (1 + \frac{2}{i})(1 - \frac{2}{i})\), we obtain '5'.

Step 4: Divide each term by the denominator

Now divide each term in the numerator by '5' to obtain the standard form of the complex number: \( \frac{8}{5} - \frac{16}{5}i\).