In Exercises 21–28, divide and express the result in standard form.


3 / 4+i

Verified step by step guidance

1. Identify the given expression to simplify: \(\frac{3}{4+i}\).
2. To express the result in standard form (a + bi), multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \$4+i\( is \)4 - i$.
3. Multiply numerator and denominator by \$4 - i$: \(\frac{3}{4+i} \times \frac{4 - i}{4 - i} = \frac{3(4 - i)}{(4+i)(4 - i)}\).
4. Simplify the denominator using the difference of squares formula: \((4+i)(4 - i) = 4^2 - (i)^2 = 16 - (-1) = 16 + 1 = 17\).
5. Expand the numerator: \$3(4 - i) = 12 - 3i$. So the expression becomes \(\frac{12 - 3i}{17}\). Finally, separate into real and imaginary parts: \(\frac{12}{17} - \frac{3}{17}i\).

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. The standard form makes it easier to perform arithmetic operations and interpret the number geometrically on the complex plane.

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 component in the denominator, resulting in a real number denominator, which is essential for simplifying complex fractions.