Simplification not so simple for me

[Titans86]

"simplification" not so simple for me...

(1+9x/4)^1/2 = 1/2(4+9X)^1/2 ?

I'm working on calculating line length using integration but my textbook keeps doing tricks like these... I can't seem to figure it out, can someone please explain this to me...

Thanks, Adam

 

[jgens]

Sure, try reexpressing the 1 so the both 1 and 9x/4 have a common denominator, then reexpress the squareroot and simplify.

 

[Defennder]

$$\sqrt{1+\frac{9}{4}x} = \sqrt{\frac{1}{4}(4+9x)} = \sqrt{\frac{1}{4}} \sqrt{4+9x}= \frac{1}{2}\sqrt{4+9x}$$

 

[Titans86]

Bah, So elementary.

Thanks guys!

 

[Dick]

(1+9x/4)=(1/4)(4+9x), ok so far? Take the square root of both sides. (1/4)^(1/2)=1/2, so (1+9x/4)^(1/2)=(1/2)*sqrt(4+9x). They just factored a 1/4 out.

 

[Titans86]

follow up question if I may... (I tried using this simplification to my demise)

I have $$\sqrt{1+\frac{1}{x^{2}}}$$

I tried he trick and got $$\left(\frac{1}{x}\right)\sqrt{x^{2}+1}$$

Yet it doesn't match... where did I go wrong?

Also, I need to be worrying about the +- when I do these things also no?

 

[Mentallic]

Well there are multiple ways of expressing these, so you'll need to show us what the textbook has provided. Your simplification is correct though.
You don't need to worry about the $$\pm$$, since that is only when you have an equation in the form x^2=y and then you make x the subject.

 

[Dick]

It should really be written (1/|x|)*sqrt(x^2+1)=sqrt(1+1/x^2). Is that what you mean by going wrong?

 

[Titans86]

I plotted it through my Ti and two different curves resulted...

edit: ah yes, the absolute would resolve it...

(my algebra is so poor...)

 

[mutton]

You don't need to worry about the $$\pm$$, since that is only when you have an equation in the form x^2=y and then you make x the subject.

But that is what is happening here. $$\sqrt{x^2} = |x|$$.