Divide.
$\frac{-5}{2-6 i}$
Write your answer as a complex number in standard form.
To divide complex numbers, we want to express the quotient in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers. Given the expression \(\frac{-5}{2-6i}\), the goal is to eliminate the imaginary component from the denominator. This is achieved by multiplying the numerator and the denominator by the complex conjugate of the denominator. For a complex number \(z = x + yi\), its complex conjugate is \(\overline{z} = x - yi\). Here, our denominator is \(2 - 6i\), so its conjugate is \(2 + 6i\).
We multiply the original fraction by \(\frac{2+6i}{2+6i}\):
\[ \frac{-5}{2-6i} \times \frac{2+6i}{2+6i} = \frac{-5(2+6i)}{(2-6i)(2+6i)} \]
Now, we expand the numerator:
\[ -5(2 + 6i) = -10 - 30i \]
Next, we expand the denominator:
\[ (2-6i)(2+6i) = 2^2 + (6)^2 = 4 + 36 = 40 \]
Thus, the expression becomes:
\[ \frac{-10 - 30i}{40} \]
Now, we split the fraction into its real and imaginary parts:
\[ \frac{-10}{40} - \frac{30i}{40} \]
Simplifying these fractions, we get:
\[ \frac{-1}{4} - \frac{3}{4}i \]
This can also be written in decimal form as \(-0.25 - 0.75i\). Both are valid representations in standard \(a + bi\) form. We will provide the fractional form as the final result.
In electrical engineering, complex numbers are used to represent impedance in AC circuits. Dividing complex numbers, such as finding the current from a given voltage and impedance, is a common task. By converting fractions like \(\frac{V}{Z}\) into the form \(a + bi\), engineers can easily separate the real part (resistance) from the imaginary part (reactance), which is essential for analyzing the power and phase shift in circuit designs.