Question $-18 i-(-5-10 i)$ 's terms as headings in the table. Then enter the product of each pail $-18 i+$ $\square$

Studdy Solution

STEP 1

What is this asking? We're asked to divide $-18i$ by $-5 - 10i$ , which is a complex number. Watch out! Remember that dividing by a complex number is like rationalizing the denominator.
We'll need its conjugate!

STEP 2

1. Rewrite the expression
2. Multiply by the conjugate
3. Simplify the numerator
4. Simplify the denominator
5. Separate the real and imaginary parts

STEP 3

Let's rewrite the division as a fraction.
This makes it easier to see what we're doing!
We have $-18i$ divided by $-5 - 10i$ , which we can write as: $\frac{-18i}{-5 - 10i}$

STEP 4

To divide by a complex number, we multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of $-5 - 10i$ is $-5 + 10i$ .
Remember, we find the conjugate by changing the sign of the imaginary part.
We're essentially multiplying by one, which doesn't change the value of the expression!
$\frac{-18i}{-5 - 10i} \cdot \frac{-5 + 10i}{-5 + 10i}$

STEP 5

Let's multiply the numerator: $-18i \cdot (-5 + 10i)$ .
Distributing $-18i$ to both terms inside the parentheses gives us: $(-18i) \cdot (-5) + (-18i) \cdot (10i)$ $90i - 180i^2$ Since $i^2 = -1$ , we can replace $i^2$ with $-1$ : $90i - 180(-1)$ $90i + 180$ So, our numerator simplifies to $180 + 90i$ .

STEP 6

Now, let's multiply the denominator: $(-5 - 10i) \cdot (-5 + 10i)$ .
Using the FOIL method (First, Outer, Inner, Last), we get: $(-5)(-5) + (-5)(10i) + (-10i)(-5) + (-10i)(10i)$ $25 - 50i + 50i - 100i^2$ The $-50i$ and $50i$ add to zero, and since $i^2 = -1$ , we have: $25 - 100(-1)$ $25 + 100$ $125$ Our denominator simplifies to 125.

STEP 7

Now we put the simplified numerator and denominator together: $\frac{180 + 90i}{125}$ We can separate this into real and imaginary parts: $\frac{180}{125} + \frac{90}{125}i$ Simplifying the fractions by dividing both numerator and denominator by their greatest common divisor (GCD), which is 5 for $\frac{180}{125}$ and 5 for $\frac{90}{125}$ : $\frac{36}{25} + \frac{18}{25}i$

STEP 8

Our final answer is $\frac{36}{25} + \frac{18}{25}i$ .

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