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 given expression: \( \frac{5}{2 - i} \). The goal is to write this expression in standard form, which means expressing it as \( a + bi \), where \( a \) and \( b \) are real numbers.
2. To eliminate the imaginary unit \( i \) from the denominator, multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \( 2 - i \) is \( 2 + i \). So multiply by \( \frac{2 + i}{2 + i} \).
3. Perform the multiplication in the numerator: \( 5 \times (2 + i) = 10 + 5i \).
4. Perform the multiplication in the denominator using the difference of squares formula: \( (2 - i)(2 + i) = 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5 \).
5. Now, write the expression as \( \frac{10 + 5i}{5} \). Then, separate the real and imaginary parts by dividing both terms in the numerator by 5: \( \frac{10}{5} + \frac{5i}{5} = 2 + i \), which is the standard form.

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing a complex number in standard form means expressing it explicitly as a sum of its real and imaginary components.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process simplifies the expression into standard form.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part from the denominator, making it a real number, which is essential for simplifying complex fractions.