Need help on simplification approach

  • #1

Homework Statement



Simplify the following:

[tex]\frac{\frac{1}{\sqrt{t^2+1}}-t^2(t^2+1)^{-3/2}}{\sqrt{\frac{1}{t^2+1}-\frac{2t^2}{(t^2+1)^2}+\frac{t^4+t^2}{(t^2+1)^3}}}[/tex]



Homework Equations



The answer in the book was:

[tex]\frac{1}{\sqrt{t^2+1}}[/tex]

I didn't believe but then I graphed both functions and sure enough they are equivalent.

The Attempt at a Solution



All I really knew to try was to factor out a 1/(t^2+1) inside the square root, but that really didn't help me see a different approach.

I'm hoping someone will have a neat trick for simplifying this mofo
 

Answers and Replies

  • #2
1,045
2
You could try getting what's under the radical to have the same denominator, and then it would be easier to see what can be factored out and pulled from the radical?
 
  • #3
Alrighty, I'll give it a shot:

[tex]\frac{(t^2+1)^2}{(t^2+1)^3}-\frac{2t^2(t^2+1)}{(t^2+1)^3}+\frac{t^4+t^2}{(t^2+1)^3}[/tex]


[tex]\frac{t^4+2t^2+1-2t^4-2t^2+t^4+t^2}{(t^2+1)^3}[/tex]

[tex]\frac{t^2+1}{(t^2+1)^3}[/tex]

[tex]\frac{1}{(t^2+1)^2}[/tex]

taking the square root, the entire equation is now:

[tex]\frac{\frac{1}{\sqrt{t^2+1}}-t^2(t^2+1)^{-3/2}}{\frac{1}{t^2+1}}[/tex]

[tex]\frac{t^2+1}{\sqrt{t^2+1}}-\frac{t^2}{\sqrt{t^2+1}}[/tex]

[tex]\frac{1}{\sqrt{t^2+1}}[/tex]

happy happy joy joy
 
  • #4
Good job. Thanks for showing the solution.
A future high school math teacher
Terry
 

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