Question 1116154
The {{{a+i*b}}} form of complex numbers is not useful for calculating products, powers, and roots.
For those purposes the polar, or trigonometric, or exponential forms are much more useful.
For example, {{{1+sqrt(3)i}}} can be written using
its absolute value or modulus, {{{r=sqrt(1^2+(sqrt(3))^2)=sqrt(1+3)=sqrt(4)=2}}} ,
and its argument {{{theta}}} such that {{{tan(theta)=sqrt(3)/1=sqrt(3)}}} .
We could say {{{theta=60^o}}} to keep it simple, or {{{theta=pi/3}}} if you must use radians.
You can write {{{1+sqrt(3)i}}} as {{{2(cos(60^o)+i*sin(60^0))}}} or as {{{e^(i*pi("1 / 3"))}}} .
All you need to remember is that when multiplying complex numbers,
absolute values multiply, and arguments add up.
 
2) So, the solutions to {{{(1+i*sqrt(3))^"1 / 3"=(2(cos(60^o)+i*sin(60^0)))^"1 / 3"}}}
are numbers of the form {{{r(cos(theta)+i*sin(theta))}}} such that
{{{(r(cos(theta)+i*sin(theta)))^3=r^3(cos(3theta)+i*sin(3theta))=2(cos(60^o)+i*sin(60^0))}}} .
That means that {{{r^3=2}}}<-->{{{r=2^"1 / 3"=root(3,2)}}} ,
and {{{3theta}}} is {{{60^o}}} or a coterminal angle.
So, the possibilities for {{{theta}}} between {{{0^o}}} and {{{360^o}}} are
{{{3theta=60^o}}} --> {{{theta=20^o}}} ,
{{{3theta=60^o+360^o=420}}} --> {{{theta=420^o/3=140^o}}} ,
and {{{3theta=60^o+720^o=780}}} --> {{{theta=780^o/3=260^o}}} .
Approximate values are
{{{matrix(3,4,theta,20^o,140^o,260^o,
cos(theta),0.9397,-0.7990,-0.1736,
sin(theta),0.3420,0.6428,-0.9848)}}} ,
so we could write the answers as
{{{2^"1 / 3"(cos(20^o)+i*sin(20^o))=approximately}}}{{{1.1839+i*0.4309}}}
{{{2^"1 / 3"(cos(140^o)+i*sin(140^o))=approximately}}}{{{-0.9652+i*0.8099}}}
{{{2^"1 / 3"(cos(260^o)+i*sin(260^o))=approximately}}}{{{0.2188-i*1.2408}}}
 
1) The solutions to {{{i^0.25=(0+i*1)^"1 / 4"=(cos(90^o)+i*sin(90^o))^"1 / 4"}}}
are four fourth roots of {{{1(cos(90^o)+i*sin(90^o))}}} with
{{{r=root(4,1)=1}}} and {{{4theta=90^o+k*2pi)}}} with {{{k="0 , 1 , 2 , or 3"}}} .
That means {{{theta=22.5^o}}}{{{or}}}{{{112.5^o}}}{{{or}}}{{{202.5^o}}}{{{or}}}{{{292.5^o}}} .
The approximate values of sine and cosine for those angles are:
{{{matrix(3,5,theta,22.5^o,112.5^o,202.5^o,292.5^o,
cos(theta),0.9239,-0.3827,-0.9239,0.3827,
sin(theta),0.3827,0.9239,-0.3827,-0.9239)}}}
So, the four complex values of {{{i^0.25}}} are
{{{cos(22.5^o)+i*sin(22.5)=approximately}}}{{{0.9239+i*0.3827}}} ,
{{{cos(112.5^o)+i*sin(112.5)=approximately}}}{{{-0.3827+i*0.9239}}}
{{{cos(22.5^o)+i*sin(22.5)=approximately}}}{{{-0.9239-i*0.3827}}} , and
{{{cos(22.5^o)+i*sin(22.5)=approximately}}}{{{0.3827-i*0.9239}}}
 
3) {{{(i-1)^0.5=sqrt(2)(cos(315^o)+i*sin(315^o))^"1 / 2"}}} has 2 values, with
{{{r=sqrt(2)^1/2=root(4,2)=approximately1.1892}}} and {{{2theta=315^o+k*360^o}}} with {{{k="0 , or 1"}}} .
That means {{{theta=(315^o+k*360^o)/2=157.5^o+k*180^o}}} with {{{k="0 , or 1"}}} , meaning {{{theta=157.5^o}}} or {{{theta=337.5^o}}} .
Approximate values for sine and cosine of those angles are
{{{matrix(3,3,theta,157.5^o,337.5^o,
cos(theta),-0.9239,0.3287,
sin(theta),0.9239,-0.3287)}}} ,
so we could write the answers as
{{{root(4,2)(cos(157.5^o)+i*sin(157.5^o))=approximately}}}{{{-1.0987+i*0.4551}}} , and
{{{root(4,2)(cos(337.5^o)+i*sin(337.5^o))=approximately}}}{{{1.0987-i*0.4551}}} .
 
4) {{{9i=9(cos(90^o)+sin(90^o))}}} and {{{1+i=sqrt(2)(cos(45^o)+i*sin(45^o))}}} ,
so their ratio can be calculated as
{{{(9i)/(1+i)}}}{{{"="}}}{{{9(cos(90^o)+sin(90^o))/(sqrt(2)(cos(45^o)+i*sin(45^o)))}}}{{{"="}}}{{{(9/sqrt(2))(cos(90^o-45^o)+i*sin(90^o-45^o))}}}{{{"="}}}{{{(9sqrt(2)/2)(cos(45^o)+i*sin(45^o))}}}
Another way to calculate that quotient is using the conjugate of the denominator:
{{{(9i)/(1+i)}}}{{{"="}}}{{{(9i/(1+i))((1-i)/(1-i))}}}{{{"="}}}{{{9i(1-i)/((1+i)(1-i))}}}{{{"="}}}{{{9(i-i^2)/(1^2-i^2)=9(i+1)/(1+1)}}}{{{"="}}}{{{9(1+i)/2=9/2+i*(9/2)}}}{{{"="}}}{{{(9sqrt(2)/2)(cos(45^o)+i*sin(45^o))}}} .
Then,
{{{((9i)/(1+i))^"1 / 6"}}}{{{"="}}}{{{(9sqrt(2)/2)^"1 / 6"(cos(theta)+i*sin(theta))}}} ,
where {{{6theta=45^o+k*360^o}}} with {{{k="0 , 1 , 2 , 3, 4, or 5"}}} .
That means
{{{theta}}}{{{"="}}}{{{(45^o+k*360^o)/6}}}{{{"="}}}{{{7.5^o+k*60^o}}} .
{{{(9sqrt(2)/2)^"1 / 6"=approximately}}}{{{1.3613}}} .
Approximate values for sine and cosine of those angles are
{{{matrix(3,7,theta,7.5^o,67.5^o,127.5^o,187.5^o,247.5^o,307.5^o,
cos(theta),0.9914,0.3827,-0.6088,-0.9914,-0.3827,0.6088,
sin(theta),0.1305,0.9239,0.7934,-0.1305,-0.9239,-0.7934)}}}
So, the six complex values of {{{((9i)/(1+i))^"1 / 6"}}} are
{{{(9sqrt(2)/2)(cos(7.5^o)+i*sin(7.5^o))=approximately}}}{{{1.3497+i*0.1777}}} ,
{{{(9sqrt(2)/2)(cos(67.5^o)+i*sin(67.5^o))=approximately}}}{{{0.5209+i*1.2577}}} ,
{{{(9sqrt(2)/2)(cos(127.5^o)+i*sin(127.5^o))=approximately}}}{{{-0.8287+i*1.0800}}} ,
{{{(9sqrt(2)/2)(cos(187.5^o)+i*sin(187.5^o))=approximately}}}{{{-1.3497-i*0.1777}}} ,
{{{(9sqrt(2)/2)(cos(247.5^o)+i*sin(247.5^o))=approximately}}}{{{-0.5209-i*1.2577}}} , and
{{{(9sqrt(2)/2)(cos(307.5^o)+i*sin(307.5^o))=approximately}}}{{{0.8287-i*1.0800}}} .