SOLUTION: What is the jth root of j? Give the exact value.

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Question 737751: What is the jth root of j? Give the exact value.
Found 3 solutions by MathLover1, Edwin McCravy, Ed Parker:
Answer by MathLover1(20849) About Me  (Show Source):
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor's answer is wrong.

Apparently, you're an electronics student who uses i for current, and 
therefore has to use j for √-1.

Sorry, I thought you wanted jj and worked that out.  I'll redo it
and put the results here later.  It's actually 4.810477381.  But
here is j%5Ej

j%5Ej%22%22=%22%22%0D%0A%0D%0Amatrix%282%2C1%2C%22%22%2Ce%5Eln%28j%5Ej%29++%29%0D%0A%0D%0A%0D%0A%0D%0A%0D%0A%22%22=%22%22e%5E%28j%2Aln%28j%29%29

Now to continue we must find an expression for ln(j)

Euler's equation is 

e%5E%28j%2Atheta%29%22%22=%22%22cos%28theta%29%2Bj%2Asin%28theta%29

Substitute theta=pi%2F2

matrix%282%2C1%2C%22%22%2Ce%5E%28j%2Aexpr%28pi%2F2%29%29%29%22%22=%22%22cos%28pi%2F2%29%2Bj%2Asin%28pi%2F2%29%22%22=%22%220%2Bj=j

Therefore since 

matrix%282%2C1%2C%22%22%2Ce%5E%28j%2Aexpr%28pi%2F2%29%29%29%22%22=%22%22j

then ln%28j%29%22%22=%22%22j%2Aexpr%28pi%2F2%29

Now we go back to where we were:
j%5Ej%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28ln%28j%5Ej%29%29%29%22%22=%22%22e%5E%28+j%2Aln%28j%29+%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28+j%2Aj%2Aexpr%28pi%2F2%29%29+%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28-1%2Aexpr%28pi%2F2%29%29%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28-pi%2F2%29%29 = 0.2078795764

Edwin

Answer by Ed Parker(21) About Me  (Show Source):
You can put this solution on YOUR website!
I'm the same person as Edwin McCravy above. 

root%28j%2Cj%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Cj%5E%281%2Fj%29%29

Let's simplify 1%2Fj

1%2Fj%22%22=%22%22expr%281%2Fj%29%2Aexpr%28j%2Fj%29%22%22=%22%22j%2F%28j%5E2%29%22%22=%22%22j%2F%28-1%29%22%22=%22%22-j

So we have

root%28j%2Cj%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Cj%5E%281%2Fj%29%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Cj%5E%281%2Fj%29%29%22%22=%22%22j%5E%28-j%29%22%22=%22%22%28j%5Ej%29%5E%28-1%29

So we find j%5Ej

j%5Ej%22%22=%22%22%0D%0A%0D%0Amatrix%282%2C1%2C%22%22%2Ce%5Eln%28j%5Ej%29++%29%0D%0A%0D%0A%22%22=%22%22e%5E%28j%2Aln%28j%29%29

Now to continue we must find an expression for ln(j)

Euler's equation is 

e%5E%28j%2Atheta%29%22%22=%22%22cos%28theta%29%2Bj%2Asin%28theta%29

Substitute theta=pi%2F2

matrix%282%2C1%2C%22%22%2Ce%5E%28j%2Aexpr%28pi%2F2%29%29%29%22%22=%22%22cos%28pi%2F2%29%2Bj%2Asin%28pi%2F2%29%22%22=%22%220%2Bj=j

Therefore since 

matrix%282%2C1%2C%22%22%2Ce%5E%28j%2Aexpr%28pi%2F2%29%29%29%22%22=%22%22j

then ln%28j%29%22%22=%22%22j%2Aexpr%28pi%2F2%29

Now we go back to j%5Ej:
j%5Ej%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28ln%28j%5Ej%29%29%29%22%22=%22%22e%5E%28+j%2Aln%28j%29+%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28+j%2Aj%2Aexpr%28pi%2F2%29%29+%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28-1%2Aexpr%28pi%2F2%29%29%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Ce%5E%28-pi%2F2%29%29 

Now we go back to 

root%28j%2Cj%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Cj%5E%281%2Fj%29%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Cj%5E%281%2Fj%29%29%22%22=%22%22matrix%282%2C1%2C%22%22%2Cj%5E%28-j%29%29%22%22=%22%22%28j%5Ej%29%5E-1%22%22=%22%22matrix%282%2C1%2C%22%22%2C%28e%5E%28-pi%2F2%29%29%5E%28-1%29%29matrix%282%2C1%2C%22%22%2Ce%5E%28pi%2F2%29%29%22%22=%22%224.810477381.

Edwin