How Can You Simplify This Complex Fractional Expression?

[scoobiedoober]

Homework Statement

Simplify the following:

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

Homework Equations

The answer in the book was:

$$\frac{1}{\sqrt{t^2+1}}$$

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

 

[QuarkCharmer]

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?

 

[scoobiedoober]

Alrighty, I'll give it a shot:

$$\frac{(t^2+1)^2}{(t^2+1)^3}-\frac{2t^2(t^2+1)}{(t^2+1)^3}+\frac{t^4+t^2}{(t^2+1)^3}$$


$$\frac{t^4+2t^2+1-2t^4-2t^2+t^4+t^2}{(t^2+1)^3}$$

$$\frac{t^2+1}{(t^2+1)^3}$$

$$\frac{1}{(t^2+1)^2}$$

taking the square root, the entire equation is now:

$$\frac{\frac{1}{\sqrt{t^2+1}}-t^2(t^2+1)^{-3/2}}{\frac{1}{t^2+1}}$$

$$\frac{t^2+1}{\sqrt{t^2+1}}-\frac{t^2}{\sqrt{t^2+1}}$$

$$\frac{1}{\sqrt{t^2+1}}$$

happy happy joy joy

 

[JsStewartFan]

Good job. Thanks for showing the solution.
A future high school math teacher
Terry