SOLUTION: I am a little confused over this so please help me solving it {{{a^p=b^q=c^r=d^s}}} and ab=cd then the value of {{{1/p+1/q-1/r-1/s}}} reduces to?

Algebra ->  Exponents -> SOLUTION: I am a little confused over this so please help me solving it {{{a^p=b^q=c^r=d^s}}} and ab=cd then the value of {{{1/p+1/q-1/r-1/s}}} reduces to?      Log On


   

Question 1045306: I am a little confused over this so please help me solving it
a%5Ep=b%5Eq=c%5Er=d%5Es and ab=cd then the value of 1%2Fp%2B1%2Fq-1%2Fr-1%2Fs reduces to?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The expression 1%2Fp%2B1%2Fq-1%2Fr-1%2Fs will be undefined if one or more of those denominators is zero.
If nothing was specified about the exponents,
we have to consider the case that one (or more) of them is zero:
If pqrs=0 , then 1%2Fp%2B1%2Fq-1%2Fr-1%2Fs is undefined.

If pqrs%3C%3E0 , then 1%2Fp%2B1%2Fq-1%2Fr-1%2Fs=highlight%280%29 :
Looking at just c%5Er=d%5Es , we can write
c%5Er=d%5Es ---> %28c%5Er%29%5E%281%2Fr%29=%28d%5Es%29%5E%281%2Fr%29 ---> c%5E%22r%2Fr%22=d%5E%22s%2Fr%22 ---> c=d%5E%22s%2Fr%22 .
Since a%5Ep=d%5Es and b%5Eq=d%5Es ,
we can also write a and b as powers of d :
a=d%5E%22s%2Fp%22 and b=d%5E%22s%2Fq%22 .
Substituting the expressions found for a , b , and c into ab=cd , we get to the solution.
a%2Ab=c%2Ad --> d%5E%22s%2Fp%22%2Ad%5E%22s%2Fq%22=d%5E%22s%2Fr%22%2Ad --> d%5E%22s%2Fp%22%2Ad%5E%22s%2Fq%22=d%5E%22s%2Fr%22%2Ad --> d%5E%28%22s%2Fp%22%2B%22s%2Fq%22%29=d%5E%28%22s%2Fr%22%2B1%29 --> s%2Fp%2Bs%2Fq=r%2Fs%2B1
And multiplkying both sides of the last equation above times 1%2Fs , we get
1%2Fp%2B1%2Fq=1%2Fr%2B1%2Fs .
So, subtracting 1%2Fr%2B1%2Fs from both sides, we get
1%2Fp%2B1%2Fq-%281%2Fr%2B1%2Fs%29=1%2Fr%2B1%2Fs-%281%2Fr%2B1%2Fs%29 --> 1%2Fp%2B1%2Fq-1%2Fr-1%2Fs=1%2Fr%2B1%2Fs-1%2Fr-1%2Fs --> highlight%281%2Fp%2B1%2Fq-1%2Fr-1%2Fs=0%29 .