Question

divide.

\\(\frac{7i}{5 - 6i}\\)

\\(\frac{7i}{5 - 6i} = \\)

(simplify your answer. type an integer or a fraction. type your answer in the form a + bi.)

Explanation:

Multiply by the conjugate

To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.
The conjugate of \(5 - 6i\) is \(5 + 6i\).

$$ \frac{7i}{5 - 6i} = \frac{7i(5 + 6i)}{(5 - 6i)(5 + 6i)} $$

Expand the numerator

Using Complex Number Multiplication:

$$ 7i(5 + 6i) = 35i + 42i^2 $$

Since \(i^2 = -1\) (using Imaginary Numbers):

$$ 35i + 42(-1) = -42 + 35i $$

Expand the denominator

Multiply the conjugate pair in the denominator:

$$ (5 - 6i)(5 + 6i) = 5^2 - (6i)^2 = 25 - 36i^2 $$

Substitute \(i^2 = -1\):

$$ 25 - 36(-1) = 25 + 36 = 61 $$

Write in standard form

Combine the simplified numerator and denominator:

$$ \frac{-42 + 35i}{61} $$

Separate into the standard form \(a + bi\):

$$ -\frac{42}{61} + \frac{35}{61}i $$

Answer:

Divide.
\(\frac{7i}{5-6i} =\) <blank>\(-\frac{42}{61} + \frac{35}{61}i\)</blank>
(Simplify your answer. Type an integer or a fraction. Type your answer in the form a + bi.)