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

## 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.

#### Attachments

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• Delta2

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Math_QED
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(1) This is no linear algebra.

(2) Please type out what's written in the images.

abs((21+7i)/(1-2i))=a√b where a=_____ and b=_____.

Office_Shredder
Staff Emeritus
Gold Member
What have you tried to do so far?

Can you calculate ##(1+2i)(1-2i)## and use that to calculate ##\frac{1}{1-2i}##?

FactChecker
Gold Member
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.

• Delta2 and Keith_McClary
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.

• Rzbs, Math100 and FactChecker
Mark44
Mentor
Thread moved to Prealgebra section, and title changed.
The original post has nothing to do with Linear Algebra.

• FactChecker
Mark44
Mentor
What have you tried to do so far?
The 2nd image in post #1 shows what the OP has tried.
Can you calculate ##(1+2i)(1-2i)## and use that to calculate ##\frac{1}{1-2i}##?
The OP did this in the 2nd image.

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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• Eclair_de_XII
Thank you so much for the hint. I was able to solve the problem with your hint.

• Delta2
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*}

• Math100 and FactChecker
FactChecker
Gold Member
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##

• Rzbs and Math100
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!