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

• chwala
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.
chwala
Gold Member
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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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

PeroK and chwala
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}##
$$=\frac{x(\sqrt{x}+y)}{y(\sqrt{x}-y)(\sqrt{x}+y)}-\frac {y\sqrt x}{y(\sqrt x-y) }=\frac{x}{y(\sqrt{x}-y)}-\frac {y\sqrt x}{y(\sqrt x-y) }=\frac{\sqrt{x}(\sqrt{x}-y)}{y(\sqrt{x}-y)}=\frac{\sqrt{x}}{y}$$

chwala
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.

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##.

SammyS and chwala
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.

I don't think the lack of a list of values in the question is relevant. Mark's point is valid.

Prof B said:
I don't think the lack of a list of values in the question is relevant. Mark's point is valid.

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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:
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.

SammyS and PeroK
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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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.

chwala
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.

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.

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...

## 1. What is the purpose of evaluating this expression?

The purpose of evaluating this expression is to simplify it and find its numerical value. This can be useful in solving equations or understanding the behavior of a system in a scientific context.

## 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.

## 3. What are some common mistakes to avoid when evaluating this expression?

Some common mistakes to avoid when evaluating this expression include forgetting to apply the order of operations, making errors when simplifying fractions, and not properly distributing negative signs or exponents.

## 4. Can this expression be evaluated for any values of x and y?

Yes, this expression can be evaluated for any values of x and y, as long as the denominator is not equal to 0. However, certain values of x and y may result in undefined or complex solutions.

## 5. How can I use the result of evaluating this expression in a scientific context?

The result of evaluating this expression can be used in various scientific applications, such as in physics equations, chemical reactions, or mathematical models. It can also help in understanding the relationship between different variables in a system.

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