In Exercises 48–57, perform the indicated operations and write the result in standard form. 6/(5+i)

Verified step by step guidance

1. Step 1: Recognize that the denominator contains a complex number (5 + i). To simplify the expression, we need to eliminate the imaginary part from the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
2. Step 2: The conjugate of (5 + i) is (5 - i). Multiply both the numerator and denominator of the fraction by (5 - i): \( \frac{6}{5+i} \cdot \frac{5-i}{5-i} \).
3. Step 3: Expand the numerator by distributing 6 across (5 - i): \( 6(5 - i) = 30 - 6i \).
4. Step 4: Expand the denominator using the difference of squares formula: \( (5+i)(5-i) = 5^2 - i^2 = 25 - (-1) = 25 + 1 = 26 \).
5. Step 5: Write the result as \( \frac{30 - 6i}{26} \). Simplify the fraction by dividing both the real and imaginary parts of the numerator by 26 to express 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, b is the imaginary part, and i is the imaginary unit defined as the square root of -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 numbers from the denominator of a fraction. This is typically 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 by a complex number. By multiplying by the conjugate, one can eliminate the imaginary part from the denominator, allowing the expression to be rewritten in standard form.