index question

x,y,z are positive numbers, if x^p=y^q=z^r, 1/p+1/q+1/r=0,
find the value of xyz.

Let x^p=y^q=z^r = C
i.e. ln x^p = ln y^q = ln z^r
=> p ln x = q ln y = r ln z = ln C
=> ln x = (ln C)/p
ln y = (ln C)/q
ln z = (ln C)/r

ln xyz
=ln x + ln y + ln z
=(ln C)/p + (ln C)/q + (ln C)/ r
= (ln C)[ 1/p+1/q+1/r ]
=ln C * 0 =0
so xyz =e^0
=1

2013-02-21 22:31:17 補充：
yes, this method works too

x^p =y^q = z^r
x = z^(r/p) and y= z^(r/q)
xyz = z^(r/p)∙z^(r/q) ∙z
= z^(r/p+r/q+r/r)
= z^0 =1