In Exercises 95–99, perform the indicated operations and write the result in standard form. 8/(1 + 2/i)

Verified step by step guidance

1. Rewrite the expression by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 + \frac{2}{i}\) is \(1 - \frac{2}{i}\).
2. Multiply the numerator and the denominator by \(1 - \frac{2}{i}\): \(\frac{8}{1 + \frac{2}{i}} \times \frac{1 - \frac{2}{i}}{1 - \frac{2}{i}}\).
3. Simplify the denominator using the difference of squares formula: \((1 + \frac{2}{i})(1 - \frac{2}{i}) = 1^2 - (\frac{2}{i})^2\).
4. Calculate \((\frac{2}{i})^2\) and simplify: \((\frac{2}{i})^2 = \frac{4}{i^2} = \frac{4}{-1} = -4\).
5. Simplify the expression: \(\frac{8(1 - \frac{2}{i})}{1 + 4}\) and write the result in standard form.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, typically expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations involving imaginary units, such as division and simplification.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, it is important to express the result in this form to clearly identify the real and imaginary components. This helps in further calculations and interpretations in various mathematical contexts.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any complex or irrational numbers from the denominator of a fraction. This is typically done by multiplying the numerator and denominator by the conjugate of the denominator. In the context of complex numbers, this process simplifies expressions and makes them easier to work with, especially when converting to standard form.