# Calculate |[(3-4i)/(5+12i)]^2|.

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Another possible reason Beowulf doesn't use a weapon is to add to his legendary reputation.  Remember, for the people of Beowulf's time, they had no written language.  Their only way to achieve immortality was to act so bravely that the bards would recite poems about their heroic deeds long after they had died.  This, too, could be one reason Beowulf is willing to fight Grendel without a weapon- he knows that if he wins, his reputation as the bravest of warriors will be cemented.

Sources:

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Beowulf plans to use the strength of his body to do battle with Grendel. It is a matter of honor . Grendel does not use any weapons, so Beowulf is going to refrain from it, as well. Beowulf possesses an honor code of heroes, and to use a weapon would indicate a weakness on his part.

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The line numbers vary depending on what translation of Beowulf you are studying, so I can't refer you to specific lines.  Beowulf says he is as brave and fearless as Grendel, so he will use no weapon against the evil monster. Beowulf also indicates that it wouldn't be a fair fight because Grendel uses no weapon besides his strength.

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To calculate | {[(3-4i)/(5+12i))]^2}|

Solution:

We di the realisation of the denominator which is in complex form

(3-4i)/(5+12i) = (3-4i)(5-12i)/(5+12i)(5-12i)

= (15 + 3*12i-4i*5-4*12i^2}/(25 -144i^2)

= {15+(36-20)i  +48}/(169) , as i^2 = -1

= {63 - 16i)}/169

Therefore ( 63 -16i) /169 =  63/169 -16/169)i

| (63/169 )- (16/169)i| = sqrt { (63/169)^2 + (-16/169)^2)

=sqrt( 63^2+16^2)/169 = sqrt4225/169 = 65/169 = 5/13.

Therefore |(3-i)/(5-12i)| = 5/13

|{(3-i)/5+12i)]^2 = (5/13)^2 = 25/169

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We'll re-write the given expression:

|[(3-4i)/(5+12i)]^2| = (|3-4i|/|5+12i|)^2

The module of a complex number is:

|z| = sqrt (a^2 + b^2), where a and b are the coefficients of the real part and the imaginary part.

z = a + b*i

Now, we'll calculate |3-4i|.

|3-4i| = sqrt (3^2 + 4^2)

|3-4i| = sqrt (9+16)

|3-4i| = sqrt 25

|3-4i| = 5

|5+12i| = sqrt (5^2 + 12^2)

|5+12i| = sqrt (25 + 144)

|5+12i| = sqrt 169

|5+12i| = 13

|[(3-4i)/(5+12i)]^2| = (5/13)^2

|[(3-4i)/(5+12i)]^2| = 25 / 169