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

[Math100]

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.

 

[member 587159]

(1) This is no linear algebra.

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

 

[Math100]

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

 

[Office_Shredder]

What have you tried to do so far?

 

[Keith_McClary]

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

 

[FactChecker]

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.

 

[Eclair_de_XII]

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.

 

[Mark44]

Thread moved to Prealgebra section, and title changed.
The original post has nothing to do with Linear Algebra.

 

[Mark44]

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.

 

[Math100]

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.

 

[Math100]

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

 

[Eclair_de_XII]

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*}

 

[FactChecker]

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.

 

[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!