SOLUTION: Hi, Please help me solve this equation involving negative rational exponents: {{{ 3z^-1-3z^(-1/2)+1= 0 }}}. Here are the steps I took to try to solve the problem: First,

Algebra ->  Equations -> SOLUTION: Hi, Please help me solve this equation involving negative rational exponents: {{{ 3z^-1-3z^(-1/2)+1= 0 }}}. Here are the steps I took to try to solve the problem: First,      Log On


   

Question 1127154: Hi,
Please help me solve this equation involving negative rational exponents: +3z%5E-1-3z%5E%28-1%2F2%29%2B1=+0+.
Here are the steps I took to try to solve the problem:
First, I used the substitution method using "u" and tried completing the square and got +12u%5E2-12u%2B4=+0+.
Then, I used the quadratic equation and got 1/ 2 plus or minus 3i sqrt( 3 ) / 2.
Last I tried solving for x^-1/2 to get the final answer, but I cannot yield the books' answer: 3 / 2 plus or minus 3i sqrt( 3 ) / 2.
Can you please help me know where I am going wrong? Thank you.
Answer by ikleyn(53937) About Me  (Show Source):
You can put this solution on YOUR website!
.
The equation is

3%2Fz - 3%2Fsqrt%28z%29 + 1 = 0.


Multiply by z both sides. You will get


3+-+3%2Asqrt%28z%29+%2B+z = 0,   or, equivalently,

z+-+3%2Asqrt%28z%29+%2B+3 = 0.


Now introduce new variable  u = sqrt%28z%29.  The last equation will take the form

u%5E2+-+3u+%2B+3 = 0.


Apply the quadratic formula


u%5B1%2C2%5D = %283+%2B-+sqrt%283%5E2+-+4%2A3%2A1%29%29%2F2 = %283+%2B-+sqrt%28-3%29%29%2F2 = %283+%2B-+i%2Asqrt%283%29%29%2F2.


Now consider separately both cases


a)  u = sqrt%28z%29 = %283+%2B+i%2Asqrt%283%29%29%2F2.  Hence, squaring, you get
 
    z = %28%283%2Bi%2Asqrt%283%29%29%2F2%29%5E2 = %289+%2B+6i%2Asqrt%283%29+-3%29%2F4 = %286%2B6i%2Asqrt%283%29%29%2F4 = %283+%2B+3i%2Asqrt%283%29%29%2F2,    


and


b)  u = sqrt%28z%29 = %283+-+i%2Asqrt%283%29%29%2F2.  Hence, squaring, you get
 
    z = %28%283-i%2Asqrt%283%29%29%2F2%29%5E2 = %289+-+6i%2Asqrt%283%29+-3%29%2F4 = %286-6i%2Asqrt%283%29%29%2F4 = %283+-+3i%2Asqrt%283%29%29%2F2.


Answer.  There are two solutions.  They are complex numbers

         %283+%2B+3i%2Asqrt%283%29%29%2F2  and  %283+-+3i%2Asqrt%283%29%29%2F2.

Solved.

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Surely, it can be done in many ways and on different paths.

I showed here one of the simplest paths/ways.

Other people may prefer other ways.