We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
174
6
avatar

Find \(\left|\left(1+2i\right)^8\right|\).

 Aug 14, 2019
 #1
avatar+2476 
+3

Remember that (a+b)2 = a+ 2ab + b2

 

My hint may be wrong I did this course a long time ago

 Aug 14, 2019
 #2
avatar+23510 
+1

Find

\(\left|\left(1+2i\right)^8\right|\).

 

\(\begin{array}{|rclrcl|} \hline && \left|\left(1+2i\right)^8\right| \\ &=& \left|\Big(\left(1+2i\right)^2\Big)^4\right| & \qquad \left(1+2i\right)^2 &=& 1+4i+4i^2 \quad | \quad i^2 = -1\\ & & & &=& 1+4i-4 \\ & & & &=& -3+4i \\ &=& \left|\left(-3+4i\right)^4\right| \\ &=& \left|\Big(\left(-3+4i\right)^2\Big)^2\right| & \qquad \left(-3+4i\right)^2 &=& 9-24i+16i^2 \quad | \quad i^2 = -1\\ & & & &=& 9-24i-16i \\ & & & &=& -7-24i \\ &=& \left|\left(-7-24i\right)^2\right| \\ &=& \left|\left(-1 \right)^2\left(7+24i\right)^2\right| \\ &=& \left| \left(7+24i\right)^2\right| & \qquad \left(7+24i\right)^2 &=& 49+336i+576i^2 \quad | \quad i^2 = -1\\ & & & &=& 49+336i-576i \\ & & & &=& -527+336i \\ &=& \left|-527+336i\right| \\ &=&\sqrt{\left(-527 \right)^2 + 336^2} \\ &=&\sqrt{390625} \\ &=&\mathbf{625} \\ \hline \end{array}\)

 

\(\mathbf{\left|\left(1+2i\right)^8\right| = 625}\)

 

laugh

 Aug 14, 2019
 #3
avatar+8829 
+3

Is the following an acceptable way to do it?

 

\(\phantom{=\quad}\left|\left(1+2i\right)^8\right|\\~\\ {=\quad}\left(\left|1+2i\right|\right)^8\\~\\ {=\quad}\left(\sqrt{1^2+2^2}\right)^8\\~\\ {=\quad}\left(\sqrt{5}\right)^8\\~\\ {=\quad}5^{\frac82}\\~\\ {=\quad}5^4\\~\\ {=\quad}625\)_

 Aug 14, 2019
 #4
avatar+23510 
+1

Hello Guest,

yes,  your way is an acceptable way to do it.

 

laugh

heureka  Aug 14, 2019
 #6
avatar
+1

Thank you heureka! smiley

Guest Aug 15, 2019
 #5
avatar+7725 
0

\(\phantom{=}|(1 + 2i)^8|\\ = |1 + 2i|^8\\ = \left(\sqrt 5\right)^8\\ = 625\)

.
 Aug 14, 2019

43 Online Users

avatar
avatar
avatar