document.write( "Question 1116154: How do I find all the values to the following?
\n" ); document.write( "1) i^0.25
\n" ); document.write( "2) (1+sqrt(3)i)^(1/3)
\n" ); document.write( "3) (i-1)^0.5
\n" ); document.write( "4) ((9i)/(1+i))^(1/6) \n" ); document.write( "

Algebra.Com's Answer #731095 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The $\"a%2Bi%2Ab\"$ form of complex numbers is not useful for calculating products, powers, and roots.
\n" ); document.write( "For those purposes the polar, or trigonometric, or exponential forms are much more useful.
\n" ); document.write( "For example, $\"1%2Bsqrt%283%29i\"$ can be written using
\n" ); document.write( "its absolute value or modulus, $\"r=sqrt%281%5E2%2B%28sqrt%283%29%29%5E2%29=sqrt%281%2B3%29=sqrt%284%29=2\"$ ,
\n" ); document.write( "and its argument $\"theta\"$ such that $\"tan%28theta%29=sqrt%283%29%2F1=sqrt%283%29\"$ .
\n" ); document.write( "We could say $\"theta=60%5Eo\"$ to keep it simple, or $\"theta=pi%2F3\"$ if you must use radians.
\n" ); document.write( "You can write $\"1%2Bsqrt%283%29i\"$ as $\"2%28cos%2860%5Eo%29%2Bi%2Asin%2860%5E0%29%29\"$ or as $\"e%5E%28i%2Api%28%221+%2F+3%22%29%29\"$ .
\n" ); document.write( "All you need to remember is that when multiplying complex numbers,
\n" ); document.write( "absolute values multiply, and arguments add up.
\n" ); document.write( "
\n" ); document.write( "2) So, the solutions to

\n" ); document.write( "are numbers of the form $\"r%28cos%28theta%29%2Bi%2Asin%28theta%29%29\"$ such that
\n" ); document.write( "

.
\n" ); document.write( "That means that $\"r%5E3=2\"$ <--> $\"r=2%5E%221+%2F+3%22=root%283%2C2%29\"$ ,
\n" ); document.write( "and $\"3theta\"$ is $\"60%5Eo\"$ or a coterminal angle.
\n" ); document.write( "So, the possibilities for $\"theta\"$ between $\"0%5Eo\"$ and $\"360%5Eo\"$ are
\n" ); document.write( " $\"3theta=60%5Eo\"$ --> $\"theta=20%5Eo\"$ ,
\n" ); document.write( " $\"3theta=60%5Eo%2B360%5Eo=420\"$ --> $\"theta=420%5Eo%2F3=140%5Eo\"$ ,
\n" ); document.write( "and $\"3theta=60%5Eo%2B720%5Eo=780\"$ --> $\"theta=780%5Eo%2F3=260%5Eo\"$ .
\n" ); document.write( "Approximate values are
\n" ); document.write( "

,
\n" ); document.write( "so we could write the answers as
\n" ); document.write( " $\"2%5E%221+%2F+3%22%28cos%2820%5Eo%29%2Bi%2Asin%2820%5Eo%29%29=approximately\"$ $\"1.1839%2Bi%2A0.4309\"$
\n" ); document.write( " $\"2%5E%221+%2F+3%22%28cos%28140%5Eo%29%2Bi%2Asin%28140%5Eo%29%29=approximately\"$ $\"-0.9652%2Bi%2A0.8099\"$
\n" ); document.write( " $\"2%5E%221+%2F+3%22%28cos%28260%5Eo%29%2Bi%2Asin%28260%5Eo%29%29=approximately\"$ $\"0.2188-i%2A1.2408\"$
\n" ); document.write( "
\n" ); document.write( "1) The solutions to

\n" ); document.write( "are four fourth roots of $\"1%28cos%2890%5Eo%29%2Bi%2Asin%2890%5Eo%29%29\"$ with
\n" ); document.write( " $\"r=root%284%2C1%29=1\"$ and $\"4theta=90%5Eo%2Bk%2A2pi%29\"$ with $\"k=%220+%2C+1+%2C+2+%2C+or+3%22\"$ .
\n" ); document.write( "That means $\"theta=22.5%5Eo\"$ $\"or\"$ $\"112.5%5Eo\"$ $\"or\"$ $\"202.5%5Eo\"$ $\"or\"$ $\"292.5%5Eo\"$ .
\n" ); document.write( "The approximate values of sine and cosine for those angles are:
\n" ); document.write( "

\n" ); document.write( "So, the four complex values of $\"i%5E0.25\"$ are
\n" ); document.write( " $\"cos%2822.5%5Eo%29%2Bi%2Asin%2822.5%29=approximately\"$ $\"0.9239%2Bi%2A0.3827\"$ ,
\n" ); document.write( " $\"cos%28112.5%5Eo%29%2Bi%2Asin%28112.5%29=approximately\"$ $\"-0.3827%2Bi%2A0.9239\"$
\n" ); document.write( " $\"cos%2822.5%5Eo%29%2Bi%2Asin%2822.5%29=approximately\"$ $\"-0.9239-i%2A0.3827\"$ , and
\n" ); document.write( " $\"cos%2822.5%5Eo%29%2Bi%2Asin%2822.5%29=approximately\"$ $\"0.3827-i%2A0.9239\"$
\n" ); document.write( "
\n" ); document.write( "3) $\"%28i-1%29%5E0.5=sqrt%282%29%28cos%28315%5Eo%29%2Bi%2Asin%28315%5Eo%29%29%5E%221+%2F+2%22\"$ has 2 values, with
\n" ); document.write( " $\"r=sqrt%282%29%5E1%2F2=root%284%2C2%29=approximately1.1892\"$ and $\"2theta=315%5Eo%2Bk%2A360%5Eo\"$ with $\"k=%220+%2C+or+1%22\"$ .
\n" ); document.write( "That means $\"theta=%28315%5Eo%2Bk%2A360%5Eo%29%2F2=157.5%5Eo%2Bk%2A180%5Eo\"$ with $\"k=%220+%2C+or+1%22\"$ , meaning $\"theta=157.5%5Eo\"$ or $\"theta=337.5%5Eo\"$ .
\n" ); document.write( "Approximate values for sine and cosine of those angles are
\n" ); document.write( "

,
\n" ); document.write( "so we could write the answers as
\n" ); document.write( " $\"root%284%2C2%29%28cos%28157.5%5Eo%29%2Bi%2Asin%28157.5%5Eo%29%29=approximately\"$ $\"-1.0987%2Bi%2A0.4551\"$ , and
\n" ); document.write( " $\"root%284%2C2%29%28cos%28337.5%5Eo%29%2Bi%2Asin%28337.5%5Eo%29%29=approximately\"$ $\"1.0987-i%2A0.4551\"$ .
\n" ); document.write( "
\n" ); document.write( "4) $\"9i=9%28cos%2890%5Eo%29%2Bsin%2890%5Eo%29%29\"$ and $\"1%2Bi=sqrt%282%29%28cos%2845%5Eo%29%2Bi%2Asin%2845%5Eo%29%29\"$ ,
\n" ); document.write( "so their ratio can be calculated as
\n" ); document.write( " $\"%289i%29%2F%281%2Bi%29\"$ $\"%22=%22\"$

$\"%22=%22\"$ $\"%289%2Fsqrt%282%29%29%28cos%2890%5Eo-45%5Eo%29%2Bi%2Asin%2890%5Eo-45%5Eo%29%29\"$ $\"%22=%22\"$ $\"%289sqrt%282%29%2F2%29%28cos%2845%5Eo%29%2Bi%2Asin%2845%5Eo%29%29\"$
\n" ); document.write( "Another way to calculate that quotient is using the conjugate of the denominator:
\n" ); document.write( " $\"%289i%29%2F%281%2Bi%29\"$ $\"%22=%22\"$ $\"%289i%2F%281%2Bi%29%29%28%281-i%29%2F%281-i%29%29\"$ $\"%22=%22\"$ $\"9i%281-i%29%2F%28%281%2Bi%29%281-i%29%29\"$ $\"%22=%22\"$ $\"9%28i-i%5E2%29%2F%281%5E2-i%5E2%29=9%28i%2B1%29%2F%281%2B1%29\"$ $\"%22=%22\"$ $\"9%281%2Bi%29%2F2=9%2F2%2Bi%2A%289%2F2%29\"$ $\"%22=%22\"$ $\"%289sqrt%282%29%2F2%29%28cos%2845%5Eo%29%2Bi%2Asin%2845%5Eo%29%29\"$ .
\n" ); document.write( "Then,
\n" ); document.write( " $\"%28%289i%29%2F%281%2Bi%29%29%5E%221+%2F+6%22\"$ $\"%22=%22\"$ $\"%289sqrt%282%29%2F2%29%5E%221+%2F+6%22%28cos%28theta%29%2Bi%2Asin%28theta%29%29\"$ ,
\n" ); document.write( "where $\"6theta=45%5Eo%2Bk%2A360%5Eo\"$ with $\"k=%220+%2C+1+%2C+2+%2C+3%2C+4%2C+or+5%22\"$ .
\n" ); document.write( "That means
\n" ); document.write( " $\"theta\"$ $\"%22=%22\"$ $\"%2845%5Eo%2Bk%2A360%5Eo%29%2F6\"$ $\"%22=%22\"$ $\"7.5%5Eo%2Bk%2A60%5Eo\"$ .
\n" ); document.write( " $\"%289sqrt%282%29%2F2%29%5E%221+%2F+6%22=approximately\"$ $\"1.3613\"$ .
\n" ); document.write( "Approximate values for sine and cosine of those angles are
\n" ); document.write( "

\n" ); document.write( "So, the six complex values of $\"%28%289i%29%2F%281%2Bi%29%29%5E%221+%2F+6%22\"$ are
\n" ); document.write( " $\"%289sqrt%282%29%2F2%29%28cos%287.5%5Eo%29%2Bi%2Asin%287.5%5Eo%29%29=approximately\"$ $\"1.3497%2Bi%2A0.1777\"$ ,
\n" ); document.write( " $\"%289sqrt%282%29%2F2%29%28cos%2867.5%5Eo%29%2Bi%2Asin%2867.5%5Eo%29%29=approximately\"$ $\"0.5209%2Bi%2A1.2577\"$ ,
\n" ); document.write( " $\"%289sqrt%282%29%2F2%29%28cos%28127.5%5Eo%29%2Bi%2Asin%28127.5%5Eo%29%29=approximately\"$ $\"-0.8287%2Bi%2A1.0800\"$ ,
\n" ); document.write( " $\"%289sqrt%282%29%2F2%29%28cos%28187.5%5Eo%29%2Bi%2Asin%28187.5%5Eo%29%29=approximately\"$ $\"-1.3497-i%2A0.1777\"$ ,
\n" ); document.write( " $\"%289sqrt%282%29%2F2%29%28cos%28247.5%5Eo%29%2Bi%2Asin%28247.5%5Eo%29%29=approximately\"$ $\"-0.5209-i%2A1.2577\"$ , and
\n" ); document.write( " $\"%289sqrt%282%29%2F2%29%28cos%28307.5%5Eo%29%2Bi%2Asin%28307.5%5Eo%29%29=approximately\"$ $\"0.8287-i%2A1.0800\"$ . \n" ); document.write( "

\n" );