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To divide radical expressions; quick question, are these equivalent?

  • Thread starter Raizy
  • Start date
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1. Homework Statement

There is no problem, ^title^, just a quick question :)

The book's answer: [tex]\frac{x\sqrt{x}-x\sqrt{2}+2\sqrt{x}-2\sqrt{2}}{x-2}} [/tex]

My answer: [tex]\frac{x+2\sqrt{x}-x-2\sqrt{2}}{x+2}[/tex]

I combined like terms in the numerator.... at least I think I did it correctly?
 

HallsofIvy

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Homework Helper
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1. Homework Statement

There is no problem, ^title^, just a quick question :)

The book's answer: [tex]\frac{x\sqrt{x}-x\sqrt{2}+2\sqrt{x}-2\sqrt{2}}{x-2}} [/tex]

My answer: [tex]\frac{x+2\sqrt{x}-x-2\sqrt{2}}{x+2}[/tex]

I combined like terms in the numerator.... at least I think I did it correctly?
What terms did you combine? There are no "like terms" in the numerator! And how did "x- 2" in the denominator become "x+ 2"?

You can write [itex]x\sqrt{x}- x\sqrt{2}= x(\sqrt{x}- \sqrt{2})[/itex] and [itex]2\sqrt{x}- 2\sqrt{2}= 2(\sqrt{x}- \sqrt{2})= 2(\sqrt{x}-\sqrt{2})[/itex] and then factor out [itex]\sqrt{x}- \sqrt{2}[/itex].
 
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a quick check is let x = something. I suggest 1 or 0.
 
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a quick check is let x = something. I suggest 1 or 0.
Not quite a good idea. Better do the factorization that Hallsofivy recommended. Putting in some arbitrary numbers can give contradictory results. For example 2x - 1 = 3 if x = 2 but 2x - 1 =/= 3 if x = 0. Even if this approach sometimes gives you the answer, there are many occasions when it won't so better not get used to it. Just take Hallsofivy's advice.
 
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I meant to compare if one function is equal to another. If you think 2x2 = 2 + x + x, you can check it by letting x = 3. After rereading the op I think I misunderstood what happened. Raizy did you just try simplifying the book's answer or is what you posted an answer you got by yourself?
 
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I meant to compare if one function is equal to another. If you think 2x2 = 2 + x + x, you can check it by letting x = 3.
That's not the way to compare if functions are equal. Except if you find that their domains are equal and than compare all values of x from their domain, which would take an infinite amount of time.

Now than I think about it again, that's the case if you need to prove that the functions are equal. But this time if we find that with at least one value of x they aren't, we've proven empirically that they're not equal, which is what the OP was asking.
 
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But this time if we find that with at least one value of x they aren't, we've proven empirically that they're not equal, which is what the OP was asking.
This is what I meant.
 
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And how did "x- 2" in the denominator become "x+ 2"?
ARGHH!! I think I screwed up copying something, and I always throw out my work sheets since I can't bare to even try and "study" off these pieces of paper with a bunch of x's and ='s and -'s and +'s and... :grumpy:

(edit)... ahH! found it. Now i need to re-do the question...

Raizy did you just try simplifying the book's answer or is what you posted an answer you got by yourself?
I'll be back :D ... I actually got the right answer but I tried to take one step further because if you would read down below, I thought the numerator would simplify even further.


Okay here it is again:

1. Homework Statement

Simplify: [tex]\frac{x+2}{\sqrt{x}+\sqrt{2}}[/tex]

The book's answer:[tex]\frac{x\sqrt{x}-x\sqrt{2}+2\sqrt{x}-2\sqrt{2}}{x-2}} [/tex]

3. The Attempt at a Solution

Step 1 - Multiply by the denominator's conjugate (Latex won't align, I'm not sure what I'm doing wrong):

=[tex]\frac{(x+2)(\sqrt{x}-\sqrt{2})}}{(\sqrt{x}+\sqrt{2})}{(\sqrt{x}-\sqrt{2})}[/tex]

Step 2 - FOIL:

=[tex]\frac{x\sqrt{x}-x\sqrt{2}+2\sqrt{x}-2\sqrt{2}}{x-2}}[/tex]

Now here's what I thought about Step 2 (which turns out to be the correct answer based on the book's answer):

The first and third term in the numerator, which are [tex]x\sqrt{x} \ \mbox{and} \ 2\sqrt{x}[/tex] would simplify to [tex]x+2\sqrt{x}[/tex]

and that

The second and fourth term in the numerator, which are [tex]-x\sqrt{2} \mbox{&} -2\sqrt{2} \ \mbox{would simplify to:} \ -x-2\sqrt{2}[/tex]
Which finally, I end up with (I have copied the down the wrong denominator in the original post, the denominator should have a negative sign) as follows:

Final answer: [tex]\frac{x+2\sqrt{x}-x-2\sqrt{2}}{x-2}[/tex]

Since my answer is apparently not equivalent, what mathematical rule have I broken?

Did I ever mention latex is a pita to use? I better get used to it :wink:
 
Last edited:
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Hi
The rule you have broken is that x[tex]\sqrt{x}[/tex] or x[tex]\sqrt{2}[/tex] are not equal to x. They can't be simplified. They are what they are. The best you can do is write it in a different form i.e. x[tex]\sqrt{x}[/tex] = x3/2

I think you have tried to take the common multiple out of the two terms as a factor but you must always use a bracket when you do this:
x[tex]\sqrt{x}[/tex] + 2[tex]\sqrt{x}[/tex] = (x+2)[tex]\sqrt{x}[/tex]
 

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