Question $\frac{7 i}{-8+6 i}$

Studdy Solution

STEP 1

1. We are given a complex fraction and need to simplify it.
2. The denominator is a complex number, and we can simplify the fraction by multiplying the numerator and the denominator by the conjugate of the denominator.

STEP 2

1. Identify the conjugate of the denominator.
2. Multiply the numerator and the denominator by the conjugate.
3. Simplify the resulting expression.

STEP 3

Identify the conjugate of the denominator $-8 + 6i$ . The conjugate is $-8 - 6i$ .

STEP 4

Multiply both the numerator and the denominator by the conjugate $-8 - 6i$ :
$\frac{7i}{-8+6i} \times \frac{-8-6i}{-8-6i} = \frac{7i(-8-6i)}{(-8+6i)(-8-6i)}$

STEP 5

Calculate the numerator:
$7i(-8-6i) = 7i \cdot (-8) + 7i \cdot (-6i) = -56i - 42i^2$
Since $i^2 = -1$ , we have:
$-42i^2 = 42$
Thus, the numerator becomes:
$42 - 56i$

STEP 6

Calculate the denominator using the difference of squares formula:
$(-8+6i)(-8-6i) = (-8)^2 - (6i)^2 = 64 - 36(-1) = 64 + 36 = 100$

STEP 7

Combine the results to simplify the expression:
$\frac{42 - 56i}{100} = \frac{42}{100} - \frac{56i}{100}$
Simplify each term:
$\frac{42}{100} = \frac{21}{50}, \quad \frac{56}{100} = \frac{28}{50}$
Thus, the simplified expression is:
$\frac{21}{50} - \frac{28}{50}i$
The simplified form of the expression is:
$\boxed{\frac{21}{50} - \frac{28}{50}i}$

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