Evaluate ##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\dfrac {\sqrt x}{\sqrt x -y}##

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In summary: It is generally a good idea to list the values of x and/or y that are not permitted. Obviously, y must not equal zero, so you don't have to explicitly list it....and then in this post you say;I get your point, which point is valid? Listing or not listing the values of ##x## and ##y?##I would list the values of x and/or y that are not permitted. Obviously, y must not equal zero, so you don't have to explicitly list it.
  • #1
chwala
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Homework Statement
Evaluate ##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\dfrac {\sqrt x}{\sqrt x -y}##
Relevant Equations
working with surds
My approach;
##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\dfrac {\sqrt x}{\sqrt x -y}##
##\dfrac {x(\sqrt x+y)}{y(x-y^2)}-\dfrac {\sqrt x}{\sqrt x -y}=\dfrac{x(\sqrt x+y)(\sqrt x-y)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x(x-y\sqrt x+y\sqrt x-y^2)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x(x-y^2)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x-y\sqrt x}{y(\sqrt x-y)}##

Now on factorization i ended up with;
##\dfrac{x-y\sqrt x}{y(\sqrt x-y)}=\dfrac{\sqrt x(\sqrt x-y)}{y(\sqrt x -y)}=\dfrac {\sqrt x}{y}## which is correct as per the textbook solution. I would appreciate an alternative approach guys.
 
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  • #2
Seems fine to me . It would make life easier to recognize and substitute $$(x-y^2) =(\sqrt x -y) (\sqrt x +y)$$ at the second step but slow and steady works
 
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  • #3
chwala said:
Homework Statement:: Evaluate ##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\dfrac {\sqrt x}{\sqrt x -y}##
Relevant Equations:: working with surds

My approach;
##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\frac {\sqrt x}{\sqrt x -y}##
$$=\frac{x(\sqrt{x}+y)}{y(\sqrt{x}-y)(\sqrt{x}+y)}-\frac {y\sqrt x}{y(\sqrt x-y) }=\frac{x}{y(\sqrt{x}-y)}-\frac {y\sqrt x}{y(\sqrt x-y) }=\frac{\sqrt{x}(\sqrt{x}-y)}{y(\sqrt{x}-y)}=\frac{\sqrt{x}}{y}$$
 
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  • #4
It is a good idea to rationalize denominators before doing anything else. That's why your teachers taught you how to do it. Try it and see.
 
  • #5
Prof B said:
It is a good idea to rationalize denominators before doing anything else.
This is generally a good idea, but not necessary in this problem, as the denominator in the second term is a factor of the denominator in the first term.

@chwala, it would be a good idea to list the values of x and/or y that are not permitted. Obviously ##y \ne 0##, so you don't have to explicitly list it, but since you canceled ##\sqrt x - y##, you should also exclude ##y = \pm \sqrt x##. Note that the original expression is undefined if ##y = 0## or if ##y = \pm \sqrt x##.
 
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  • #6
Mark44 said:
This is generally a good idea, but not necessary in this problem, as the denominator in the second term is a factor of the denominator in the first term.

@chwala, it would be a good idea to list the values of x and/or y that are not permitted. Obviously ##y \ne 0##, so you don't have to explicitly list it, but since you canceled ##\sqrt x - y##, you should also exclude ##y = \pm \sqrt x##. Note that the original expression is undefined if ##y = 0## or if ##y = \pm \sqrt x##.
I posted as it is... on textbook. ...no values to list.

1653083612850.png
 
  • #7
I don't think the lack of a list of values in the question is relevant. Mark's point is valid.
 
  • #8
Prof B said:
I don't think the lack of a list of values in the question is relevant. Mark's point is valid.
You're contradicting yourself...Read @Mark44 comment i.e post ##5## again...
 
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  • #9
Prof B said:
I don't think the lack of a list of values in the question is relevant. Mark's point is valid.

chwala said:
You're contradicting yourself...
No he isn't. The list of values here is the set of numbers that must be excluded from consideration. Sometimes it's shown explicitly, and sometimes the list is implicit. For the problem at hand, the original expression is defined for all x, y such that ##x \ge 0##, ##y \ne 0##, and ##y \ne \pm \sqrt x##.

For the simplified expression to remain equal to the starting expression, we have to be dealing with the same limited domains on x and y. For the resulting expression, ##\frac{\sqrt x}y##, it's obvious that ##x \ge 0## and ##y \ne 0##, so it's not absolutely necessary to list these values. However, if ##y = \sqrt x##, the ending expression has a value of 1, but the expression we started with is undefined. That was my point, that we should list any values x and that can't be used because the starting and ending expressions won't be equal.
 
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  • #10
Mark44 said:
No he isn't. The list of values here is the set of numbers that must be excluded from consideration. Sometimes it's shown explicitly, and sometimes the list is implicit. For the problem at hand, the original expression is defined for all x, y such that ##x \ge 0##, ##y \ne 0##, and ##y \ne \pm \sqrt x##.

For the simplified expression to remain equal to the starting expression, we have to be dealing with the same limited domains on x and y. For the resulting expression, ##\frac{\sqrt x}y##, it's obvious that ##x \ge 0## and ##y \ne 0##, so it's not absolutely necessary to list these values. However, if ##y = \sqrt x##, the ending expression has a value of 1, but the expression we started with is undefined. That was my point, that we should list any values x and that can't be used because the starting and ending expressions won't be equal.
I get your point, which point is valid? Listing or not listing the values of ##x## and ##y?## You were of the idea that it's relevant to list the permitted values of ##x## and/or ##y##, which in that case makes your point valid. That's why I indicated the response as being a contradiction...unless I am interpreting it wrongly...I may be wrong, just by looking at your post ##9## cheers.
 
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  • #11
chwala said:
I get your point, which point is valid? Listing or not listing the values of x and y?
My point is that you should list any values that must be excluded but aren't obvious in your final answer. In this case, ##y \ne \sqrt x##. This is really what I've been saying all along.
 
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  • #12
Mark44 said:
My point is that you should list any values that must be excluded but aren't obvious in your final answer. In this case, ##y \ne \sqrt x##. This is really what I've been saying all along.
Agreed! That's why I mentioned that the response that is, post ##8## was a contradiction. The response indicates,
1. It's not relevant to list the values.
2. States your point is valid.
Unless my understanding of English is wrong there, I stand corrected.
Clearly ##1## contradicts ##2##.
 
  • #13
I don't think that @Prof B was completely clear in what he wrote, but I understood it.
chwala said:
1. It's not relevant to list the values.
I believe he meant listing the prohibited values in the problem statement.
chwala said:
2. States your point is valid.
Which meant listing the prohibited values in the solution.
 
  • #14
Mark44 said:
I don't think that @Prof B was completely clear in what he wrote, but I understood it.

I believe he meant listing the prohibited values in the problem statement.

Which meant listing the prohibited values in the solution.
Ok noted, thanks...
 

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2. How do I approach evaluating this expression?

To evaluate this expression, you can start by simplifying the numerator and denominator separately. Then, you can combine like terms and use algebraic rules to manipulate the expression and simplify it further.

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