Find the quotient. Express your answer in standard form.−5+3i−7−4i\frac{-5+3i}{-7-4i}

23 65 − 41 65 i \frac{23}{65}-\frac{41}{65}i 65 23 − 65 41 i

Find the quotient. Express your answer in standard form. − 5 + 3 i − 7 − 4 i \frac{-5+3i}{-7-4i}

In this problem, we want to find the quotient of these two complex numbers and express our answer in standard form and here we have 6+i divided by 4 minuteus 2 I. Now let's take a look at our steps for dividing complex numbers. Now our first step is going to be to multiply both the top and the bottom by the complex conjugate of the bottom. And since my denominator here is 4 minuteus 2 I, the conjugate here, the complex conjugate, is going to be 4+2i. So I wanna go ahead and multiply both the top and the bottom by this 4+2i. So let's go ahead and multiply these out. So in the numerator here using foil, I can go ahead and multiply my first term, 6 times 4, which will give me 24. And then my outside terms, 6 times positive 2 I will give me positive 12 I. Then my inside terms, which will give me a 4 I, and my last terms, which will give me I times 2 I, which is positive 2 I squared. And then that's all divided by 4 minus 2 I times 4+2i. Now we can foil this out, but if you remember the trick here that if we take a complex number and multiply it by its conjugate, this is just going to become a squared plus b squared, which in this case is 4 squared plus 2 squared. We know that 4 squared is 16 and 2 squared is 4, so my denominator here is just this 16+4, which we can go ahead and simplify down to 20. So this is the fraction that I currently have. Let's go ahead and simplify this a little bit more. So we know that I squared is just negative one. So this whole term right here, 2i squared, is really just 2 times negative one. So let's go ahead and rewrite this with a couple of simplifications. So I have 24 +12i plus 4i and then minus 2, which is from that 2 times negative 1. Then all of this is divided by 20. Okay. So step 1 is done. Now we need to go ahead and combine all of our like terms for step 2. So in my numerator, I have a 24 and a negative 2 that can get combined, and I have a 12 and a 12 I and a 4 I that can be combined as well. So going ahead and combining those like terms, 24 minus 2 is going to give me 22, and then a +12i+4i will give me 16i. All of that still over 20. Okay. All my like terms are combined. Step 2 is done. Step 3 is then to expand my fraction into the real and imaginary parts. So let's go ahead and split our fraction here. So if I split the numerator, I get 22 over 20 plus 16 over 20 times I. So we've expanded that out, and then we have one final step here, which is to go ahead and simplify our fractions to their lowest terms. So this 22 over 20, 22 and 20 are both divisible by 2, so this is going to simplify to 11 over 10 if I go ahead and simplify that. And then 16 over 20, 16 and 20 are both divisible by 4, so this will simplify to 4 5ths I. So we're done and this is our solution here. I'll see you in the next video.

Find the quotient. Express your answer in standard form. − 5 + 3 i − 7 − 4 i \frac{-5+3i}{-7-4i}

23 65 − 41 65 i \frac{23}{65}-\frac{41}{65}i 65 23 − 65 41 i

3 5 + 4 5 i \frac35+\frac45i 5 3 + 5 4 i

9. Roots, Radicals, and Complex Numbers

Multiplying and Dividing Complex Numbers

Intermediate Algebra

9. Roots, Radicals, and Complex Numbers

Multiplying and Dividing Complex Numbers

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Multiple Choice

Find the quotient. Express your answer in standard form.
$\frac{- 5 + 3 i}{- 7 - 4 i}$

$\frac35+\frac45i$

$18i$

$23-41i$

$\frac{23}{65}-\frac{41}{65}i$

Verified step by step guidance

Identify the given complex fraction: $\frac{-5 + 3i}{-7 - 4i}$. Our goal is to express this quotient in standard form, which is $a + bi$, where $a$ and $b$ are real numbers.

To simplify the division of complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $-7 - 4i$ is $-7 + 4i$. So, multiply numerator and denominator by $-7 + 4i$:

\[\frac{-5 + 3i}{-7 - 4i} \times \frac{-7 + 4i}{-7 + 4i}\]

Use the distributive property (FOIL) to expand both the numerator and the denominator:

- Numerator: $(-5 + 3i)(-7 + 4i)$

- Denominator: $(-7 - 4i)(-7 + 4i)$

Calculate the denominator first, recognizing it as a difference of squares: $a^2 - b^2$, where $a = -7$ and $b = 4i$. This will result in a real number.

Expand the numerator by multiplying each term carefully, remembering that $i^2 = -1$. Then combine like terms to write the numerator in the form $A + Bi$.

Finally, write the quotient as $\frac{A}{D} + \frac{B}{D}i$, where $D$ is the real number from the denominator. This is the standard form $a + bi$.

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