## Simplifying a messy expression

Quadratic equations and inequalities, variation equations, function notation, systems of equations, etc.

### Simplifying a messy expression

Need to combine and simplify the following expression

1/(7-4*SQRT(3)) - 1/(4*SQRT(3)-SQRT(47)) + 1/(SQRT(47) - SQRT(46)) - 1/(SQRT(46)-3*SQRT(5)) +

1/(3*SQRT(5)-2*SQRT(11)) ......... - 1/(SQRT(5) - 2)
Chasyesker

Posts: 12
Joined: Wed Sep 25, 2013 7:04 pm

Chasyesker wrote:Need to combine and simplify the following expression

1/(7-4*SQRT(3)) - 1/(4*SQRT(3)-SQRT(47)) + 1/(SQRT(47) - SQRT(46)) - 1/(SQRT(46)-3*SQRT(5)) +

1/(3*SQRT(5)-2*SQRT(11)) ......... - 1/(SQRT(5) - 2)

Does the above mean the following?

$\frac{1}{7\, -\, 4 \sqrt{3}}\, -\, \frac{1}{4 \sqrt{3}\, -\, \sqrt{47}}\, +\, \frac{1}{\sqrt{47}\, -\, \sqrt{46}}\, -\, \frac{1}{\sqrt{46}\, -\, 3 \sqrt{5}}\, +\, \frac{1}{3 \sqrt{5}\, -\, 2 \sqrt{11}}\, ... \, -\, \frac{1}{\sqrt{5}\, -\, 2}$

Also, what is included in the ellipsis (the "dot dot dot") near the end? Is the pattern as follows (so, in particular, the sign on your last fraction should have been a "plus")?

$\frac{(-1)^{48}}{\sqrt{49}\, -\, \sqrt{48}}\, +\, \frac{(-1)^{47}}{\sqrt{48}\, -\, \sqrt{47}}\, +\, \frac{(-1)^{46}}{\sqrt{47}\, -\, sqrt{46}}\, +\, ... \, +\, \frac{(-1)^5}{\sqrt{6}\, -\, \sqrt{5}}\, +\, \frac{(-1)^4}{\sqrt{5}\, -\, \sqrt{4}}$

When you reply, please show your work so far, and provide any rules,theorems, or other results from your class which you feel might be relevant. Thank you!

stapel_eliz

Posts: 1803
Joined: Mon Dec 08, 2008 4:22 pm

### Re: Simplifying a messy expression

That's correct. I don't have access to the radical symbols so sorry for the messiness. I got as far as figuring out the pattern. The problem as stated has a negative sign for the last term but I think you are correct that it s/b +. The assigment sheets have ben known to be a bit sloppy when it comes to exactness.
Chasyesker

Posts: 12
Joined: Wed Sep 25, 2013 7:04 pm

### Re: Simplifying a messy expression

This might help a little:

$\frac{1}{\sqrt{n+1}-\sqrt{n}}\, =\, \frac{1}{\sqrt{n+1}-\sqrt{n}}\times \frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}\, =\, \frac{\sqrt{n+1}+\sqrt{n}}{(n+1)-(n)}\, =\, {\sqrt{n+1}+\sqrt{n}}$

$(-1)^n ({\sqrt{n+1}+\sqrt{n}})$

The fractions start at n=48 and go to n=4.

$(\sqrt{49}+\sqrt{48})\, -\, (\sqrt{48}+\sqrt{47})\, +\, (\sqrt{47}+\sqrt{46})\, -\, ...\, -\, (\sqrt{6}+\sqrt{5})\, +\, (\sqrt{5}+\sqrt{4})$

$\sqrt{49}\, +\, (\sqrt{48}-\sqrt{48})\, +\, (-\sqrt{47}+\sqrt{47})\, +\,... \, +\, (-\sqrt{5}+\sqrt{5})\, +\, \sqrt{4}$

So it comes down to 7+2=9. That seemed too easy, though, so I checked in Wolfram Alpha, and got the same thing, so I guess it's right!

maggiemagnet

Posts: 297
Joined: Mon Dec 08, 2008 12:32 am

### Re: Simplifying a messy expression

Yep,

That's what I eventually got and like you it seemed to easy but I guess they are not always hard. Thanks for spotting the wrong sign in the problem on the last term. That's how the sign was shown on the handout so I am going to assume it was a typo. I wrote it all out by hand and it should be "+". Hopefully I won't have to argue with the teacher.

Thanks again.
Chasyesker

Posts: 12
Joined: Wed Sep 25, 2013 7:04 pm