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 divide these complex numbers and then express our answer in standard form. And here I have negative 5+3i divided by negative 7 minus 4i. So So let's go ahead and start with step 1, which is gonna be to multiply both the top and the bottom by the complex conjugate of the bottom. So since my denominator here is negative 7 minus 4 I, the complex conjugate of that is going to be negative 7+4i because we're reversing the sign of that imaginary part, and we wanna multiply both the top and the bottom by this negative 7+4i. So let's go ahead and foil this out with our numerator. So starting with my first term, I have negative 5 times negative 7, it's going to give me positive 35. And then my outside term, negative 5 times 4 I, is going to give me negative 20 I. Then I have my inside term, 3 I times negative 7, it's going to give me negative 21i. And then my last term, 3i times 4i will give me positive 12i squared. Okay. So looking at my denominator here, if we remember our shortcut that this just becomes a squared plus b squared. Now you can always foil this out, of course, but knowing that these 2 multiplied together just become a squared plus b squared, this is gonna be negative 7 squared plus negative 4 squared, and negative 7 squared is positive 49, plus negative 4 squared is positive 16, and these 2 added together is going to give me 65. So my denominator here is just 65. So we've completed step 1. Now step 2 is to go ahead and combine like terms. And remember we can also simplify this I squared here into a negative one, so this is really just 12 times negative one. Let's go ahead and rewrite this simplifying a little bit. So I have 35 minus 20 I minus 21 I minus 12 because this 12 times negative one all divided by 65. Okay. Now we can go ahead and combine like terms. So on the numerator here, I have 35 and I have a negative 12 and then I also have negative 20i and negative 21i. So let's go ahead and combine those like terms. Now 35 minus 12 is going to give me 23 minus 20 I minus 21 I is negative 41 I. And this is all divided by 65. So step 2 is done as well. We can go ahead and expand our fraction into the real and imaginary parts. So splitting that numerator there, I am left with 23 over 65 minus 41 over 65 I. Okay. So step 3 is done as well. We can simplify fractions to lowest terms, and these are actually already in their lowest terms. Now I know that these fractions look a little bit crazy but we're completely done and this is our solution. Let me know if you have any questions and I'll see you in the next one.

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

11. Roots, Radicals, and Complex Numbers

Multiplying and Dividing Complex Numbers

Beginning & Intermediate Algebra

11. 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 fraction, 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: $(-5)(-7) + (-5)(4i) + (3i)(-7) + (3i)(4i)$

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

Simplify the denominator using the fact that $i^2 = -1$, and then combine like terms in both numerator and denominator. Finally, write the result as a complex number in the form $a + bi$ by dividing the real and imaginary parts of the numerator by the simplified denominator.

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