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

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Identify the given expression: \(\frac{6}{5+i}\). The goal is to write this expression in standard form, which means expressing it as \(a + bi\), where \(a\) and \(b\) are real numbers.

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 \(5 + i\) is \(5 - i\).

Multiply numerator and denominator by \(5 - i\): \(\frac{6}{5+i} \times \frac{5 - i}{5 - i} = \frac{6(5 - i)}{(5+i)(5 - i)}\).

Expand the numerator: \(6(5 - i) = 30 - 6i\). Expand the denominator using the difference of squares formula: \((5+i)(5 - i) = 5^2 - i^2 = 25 - (-1) = 25 + 1 = 26\).

Write the expression as \(\frac{30 - 6i}{26}\), then separate into real and imaginary parts: \(\frac{30}{26} - \frac{6}{26}i\). Simplify the fractions if possible to get the final standard form \(a + bi\).

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주요 개념

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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.

Rationalizing the Denominator

When dividing by a complex number, the denominator is often rationalized by multiplying numerator and denominator by the complex conjugate of the denominator. This process eliminates the imaginary part from the denominator, simplifying the expression.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, specifically a^2 + b^2, which is useful for simplifying division involving complex numbers.