With what exponent the power of any number becomes 0? Like we know that (any number)^0=1,so what will be the value of x in (any number)^x=0?

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Answer 1 · 2016-07-29T17:14:23

See below

Let #z# be a complex number with structure

#z = rho e^{i phi}# with #rho > 0, rho in RR# and #phi = arg(z)#

we can ask this question. For what values of #n in RR# occurs

#z^n=0# ?

Developping a little more

#z^n=rho^n e^{i n phi} = 0->e^{i n phi} = 0#

because by hypothese

#rho > 0#.

So using Moivre's identity

#e^{i n phi} =cos( n phi)+i sin( n phi)# then

#z^n=0->cos(n phi) +i sin( n phi)= 0-> n phi = pi + 2k pi, k=0,pm1,pm2,pm3,cdots#

Finally, for

#n = ( pi + 2k pi)/phi, k = 0,pm1,pm2,pm3,cdots#

we get

#z^n=0#