In Exercises 1–3, perform the indicated operations and write the result in standard form.5 / 2−i

Verified step by step guidance

1. Identify the expression: \( \frac{5}{2-i} \).
2. To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2-i\) is \(2+i\).
3. Perform the multiplication: \( \frac{5}{2-i} \times \frac{2+i}{2+i} = \frac{5(2+i)}{(2-i)(2+i)} \).
4. Simplify the denominator using the difference of squares formula: \((2-i)(2+i) = 2^2 - i^2 = 4 - (-1) = 5\).
5. Distribute in the numerator: \(5(2+i) = 10 + 5i\). Now, write the expression in standard form: \( \frac{10 + 5i}{5} = 2 + i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part and b is the coefficient of the imaginary unit i (where i² = -1). Understanding complex numbers is essential for performing operations involving them, such as addition, subtraction, multiplication, and division.

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 format is crucial for further mathematical analysis and applications.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any imaginary unit i from the denominator of a fraction. This is typically done by multiplying both the numerator and the denominator by the conjugate of the denominator. This process simplifies the expression and allows for easier manipulation and interpretation of complex numbers.