Question 1050430
.
Simplify (i+1)^3200-(i-1)^3200
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
i + 1 = {{{sqrt(2)*((sqrt(2)/2) + i*(sqrt(2)/2))}}} = {{{sqrt(2)*(cos(pi/4) + i*sin(pi/4))}}}.

Now apply de Moivre's formula:  {{{(r*(cos(alpha) + i*sin(alpha)))^n}}} = {{{r^n*(cos(n*alpha) + i*sin(n*alpha))}}}.

You will get

{{{(i+1)^3200}}} = {{{(sqrt(2))^3200*(cos(3200*(pi/4)) + i*sin(3200*(pi/4)))}}} = {{{2^1600*(cos(800*pi) + i*sin(800*pi))}}} = {{{2^1600}}}.    (1)



Similarly, 

i - 1 = {{{sqrt(2)*((sqrt(2)/2) - i*(sqrt(2)/2))}}} = {{{sqrt(2)*(cos(pi/4) - i*sin(pi/4))}}}.

{{{(i-1)^3200}}} = {{{(sqrt(2))^3200*(cos(3200*(pi/4)) - i*sin(3200*(pi/4)))}}} = {{{2^1600*(cos(800*pi) - i*sin(800*pi))}}} = {{{2^1600}}}.    (2)



Combining (1) and (2), you will get

(i+1)^3200-(i-1)^3200 = {{{2^1600}}} - {{{2^1600}}} = 0.

<U>Answer</U>.  (i+1)^3200-(i-1)^3200 = 0.
</pre>

There is a bunch of my lessons on complex numbers

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetical operations on them</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson>Addition and subtraction of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Multiplication-and-division-of-complex-numbers-in-complex-plane-.lesson>Multiplication and division of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Raising-a-complex-number-to-an-integer-power.lesson>Raising a complex number to an integer power</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-root-of-a-complex-number.lesson>How to take a root of a complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-real-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with real coefficients on complex domain</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-square-root-of-a-complex-number.lesson>How to take a square root of a complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-complex-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with complex coefficients on complex domain</A>

in this site.