Help me with this Algebra problem please (quotient of complex numbers)

In summary: I will make sure to show more detail in my work in the future. After factoring out as much as possible and using properties of the modulus, I was able to simplify the problem to 7√2.
  • #1
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Homework Statement
Help me with this Algebra problem?
Relevant Equations
None.
Below is the problem and the correct answer for this algebra problem is 7√2. But I cannot get to the correct answer.
 

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  • #2
(1) This is no linear algebra.

(2) Please type out what's written in the images.
 
  • #3
abs((21+7i)/(1-2i))=a√b where a=_____ and b=_____.
 
  • #4
What have you tried to do so far?
 
  • #5
Can you calculate ##(1+2i)(1-2i)## and use that to calculate ##\frac{1}{1-2i}##?
 
  • #6
Your work is correct as far as it goes (although it is not the easiest way to get the answer). To continue, apply the Pythagorean theorem to that result to get the magnitude of the complex number.

PS. I think it would be easier to start with the fact that |z1/z2| = |z1| / |z2| and apply the Pythagorean theorem to both the numerator and the denominator separately.
 
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  • #7
Reminder: ##|x+yi|=\sqrt{x^2+y^2}##.

You already found ##x+yi##. The problem asks you for ##|x+yi|##, so the next step should be straight-forward.
 
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  • #8
Thread moved to Prealgebra section, and title changed.
The original post has nothing to do with Linear Algebra.
 
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  • #9
Office_Shredder said:
What have you tried to do so far?
The 2nd image in post #1 shows what the OP has tried.
Keith_McClary said:
Can you calculate ##(1+2i)(1-2i)## and use that to calculate ##\frac{1}{1-2i}##?
The OP did this in the 2nd image.
 
  • #10
Eclair_de_XII said:
Reminder: ##|x+yi|=\sqrt{x^2+y^2}##.

You already found ##x+yi##. The problem asks you for ##|x+yi|##, so the next step should be straight-forward.
 

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  • #11
Thank you so much for the hint. I was able to solve the problem with your hint.
 
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  • #12
Math100 said:
Thank you so much for the hint. I was able to solve the problem with your hint.

Sorry for the late reply. Your work looks alright. But it might help, in the future, to show a little bit more detail than this. The answer would be clearer to the grader if you had written something like:

\begin{align*}
\sqrt{\frac{49}{25}+\frac{2401}{25}}&=\sqrt{\frac{1}{25}(49+2401)}\\
&=\sqrt{\frac{1}{25}(2450)}\\
&=\sqrt{\frac{1}{25}\cdot (2\cdot 25\cdot 49)}\\
&=\sqrt{(\frac{1}{25}\cdot 25)(2\cdot 49)}\\
&=\sqrt{2\cdot 49}\\
&=\sqrt{2\cdot 7^2}\\
&=\sqrt{2}\cdot \sqrt{7^2}\\
&=7\sqrt{2}
\end{align*}
 
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  • #13
A simplification is to factor out as much as you can as soon as you can and use the basic properties of the modulus:
|(21+7i)/(1+2i)|
=|21+7i|/|1+2i|
= 7*|3+i|/|1+2i|
= 7*##\sqrt {10}##/##\sqrt 5##
= 7*##\sqrt 5 ## * ##\sqrt 2##/##\sqrt 5##
= 7##\sqrt 2##

EDIT: PS. You can often do these things in homework and exam problems that are "rigged" (for one thing, so that the teacher is sure that he has the right answer.) It is not so common in real-world problems.
 
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  • #14
Eclair_de_XII said:
Sorry for the late reply. Your work looks alright. But it might help, in the future, to show a little bit more detail than this. The answer would be clearer to the grader if you had written something like:

\begin{align*}
\sqrt{\frac{49}{25}+\frac{2401}{25}}&=\sqrt{\frac{1}{25}(49+2401)}\\
&=\sqrt{\frac{1}{25}(2450)}\\
&=\sqrt{\frac{1}{25}\cdot (2\cdot 25\cdot 49)}\\
&=\sqrt{(\frac{1}{25}\cdot 25)(2\cdot 49)}\\
&=\sqrt{2\cdot 49}\\
&=\sqrt{2\cdot 7^2}\\
&=\sqrt{2}\cdot \sqrt{7^2}\\
&=7\sqrt{2}
\end{align*}

Thank you so much for the help!
 

1. What are complex numbers?

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part with i being the square root of -1.

2. How do you find the quotient of two complex numbers?

To find the quotient of two complex numbers, you need to first multiply the numerator and denominator by the complex conjugate of the denominator. Then, you can simplify the resulting expression to get the final quotient.

3. What is the complex conjugate of a complex number?

The complex conjugate of a complex number is the same number with the sign of the imaginary part changed. For example, the complex conjugate of 3 + 4i is 3 - 4i.

4. Can you provide an example of finding the quotient of complex numbers?

Sure, let's say we want to find the quotient of (2 + 3i) / (1 + 2i). We would first multiply the numerator and denominator by the complex conjugate of the denominator, which is (1 - 2i). This gives us (2 + 3i)(1 - 2i) / (1 + 2i)(1 - 2i). Simplifying this, we get (4 + 7i) / 5, which is our final quotient.

5. Are there any special rules for dividing complex numbers?

Yes, when dividing complex numbers, you need to remember to multiply by the complex conjugate of the denominator. This ensures that the denominator becomes a real number, making it easier to simplify the expression.

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