In Exercises 1–10, perform the indicated operations and write the result in standard form.6 / 5+i

Verified step by step guidance

1. Multiply the numerator and the denominator by the conjugate of the denominator, which is \(5 - i\).
2. Write the expression as \(\frac{6}{5+i} \times \frac{5-i}{5-i}\).
3. Distribute in the numerator: \(6 \times (5 - i)\).
4. Use the difference of squares formula in the denominator: \((5+i)(5-i) = 5^2 - i^2\).
5. Simplify the expression to write the result in standard form \(a + bi\).

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. To express a complex number in standard form, it is often necessary to eliminate any imaginary unit from the denominator, which can be achieved by multiplying the numerator and denominator by the conjugate of the denominator.

Conjugate of a Complex Number

The conjugate of a complex number a + bi is a - bi. The conjugate is useful in simplifying expressions involving complex numbers, particularly when dividing them. By multiplying by the conjugate, one can eliminate the imaginary part from the denominator, allowing for the expression to be rewritten in standard form.