SOLUTION: Hi - In a recent demonstration, somebody went from this (square root of (1 + x-squared))/x to this square root of (1 + 1/x-squared) They're both obviously eq

Algebra ->  Algebra  -> Square-cubic-other-roots -> SOLUTION: Hi - In a recent demonstration, somebody went from this (square root of (1 + x-squared))/x to this square root of (1 + 1/x-squared) They're both obviously eq      Log On

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Question 107071: Hi - In a recent demonstration, somebody went from this

(square root of (1 + x-squared))/x
to this

square root of (1 + 1/x-squared)

They're both obviously equivalent, as both result in the same answer for values of x. Can you help me to understand how this manipulation was performed, and how these two forms relate? Thanks!


Answer by HyperBrain(694) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29
THIS IS INTERESTING!!! Thanks! I love this type of problem.
Note that the equation above is the same as
sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29%2F%281%29
By cross multiplication,
x%28sqrt%281%2B%281%2Fx%5E2%29%29%29=1%28sqrt%281%2Bx%5E2%29%29
x%28sqrt%281%2B%281%2Fx%5E2%29%29%29=sqrt%281%2Bx%5E2%29
If you square both sides,
%28x%28sqrt%281%2B%281%2Fx%5E2%29%29%29%29%5E2=%28sqrt%281%2Bx%5E2%29%29%5E2
x%5E2%281%2B1%2Fx%5E2%29%29=%281%2Bx%5E2%29=1%2Bx%5E2
x%5E2%281%29+%2B+x%5E2%281%2Fx%5E2%29=1%2Bx%5E2
x%5E2%2B+%28x%5E2%2Fx%5E2%29=1%2Bx%5E2
x%5E2%2B1=1%2Bx%5E2-------------we're getting close! watch the next move or
maybe you can do it. hint:it IS easy
The final move....
x%5E2%2B1=x%5E2%2B1--------------can you guess what I did? From 1%2Bx%5E2, I
just interchanged 1 and x%5E2

Therefore, sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29 is proven to be true.

Power up,
HyperBrain!