Problem 77 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The operation results in the standard form of the complex number \(1.6 - 3.2i\)

Step 1: Simplifying the Denominator

Multiply the whole fraction by the conjugate of the denominator which is \(1-2i\). This is done to get the denominator to a real number form. Hence, \(\frac{8}{1+\frac{2}{i}} \times \frac{1-2i}{1-2i} = \frac{8(1- 2i)}{(1+2i)(1-2i)}\).

Step 2: Simplify the Result

Now, simplify the expression. The denominator simplifies from the difference of two squares formula, and the numerator is simply distributed to get \(8 - 16i\). Hence, we get \(\frac{8 - 16i}{1+4} = \frac{8 - 16i}{5}\). The result is now almost in standard form.

Step 3: Write in Standard Form

Divide each real and imaginary part by 5 to reach the standard form of a complex number. Hence, the final answer in standard form is \(\frac{8}{5} - \frac{16}{5}i\) or \(1.6 - 3.2i\).