Hi. I've almost finished my assignment but there are three questions that I just can't figure out. If someone could provide the steps to finding the answer, I would really appreciate it. (One thing that really confuses me is when a part of the equation looks like this: x^2(2x-1)^{1/2}. Are we allowed to multiply the first part without doing anything with the exponent?) Anyway, here are the questions:

#1: [y^{3/8} (y^{5/8} - y^{13/8})] / [y^{1/2} (y^{1/2} - y^{-1/2})]

#2: [(a^{1/2} - a^{-1/2})^{2} +4]^{1/2}

#3: x^{2} (2x - 1)^{-1/2} + 2x (2x - 1)^{1/2}

Do you mean this?



and using the distributive law?
This you cannot because the terms should be of the same power. For example, if it were:



Then, you can put it this way:



1.



Expand the numerator and the denominator, leave the powers as they are, don't simplify now.



Now, use the product rule, that is



So, this becomes:



Simplify the fractions;





Which becomes:



Now simplify further by factorising the numerator by -y:







Can you try the others now and post what you get? :)

2.



Expand the quadratic first.
Convert everything into a single fraction.
Factorise the numerator.
Then, insert the power (1/2) into the brackets.
Now, you can separate the improper fraction into two terms.

3.



Factorise first. You can factorise x and (2x-1)^(-1/2).
Remember that



Simplify the terms in brackets and put the whole thing into a more 'aesthetic' way.

Oh my goodness! I have a big 'note-to-self' saying 'DON'T forget about factoring' and that's just what I didn't do. I forgot all about it. Anyway, here's what I've done for #2:

#2: [(a^{1/2} - a^{-1/2})^{2} +4]^{1/2}

[((a^{1/2}-a^{-1/2})(a^{1/2}-a^{-1/2}))+4]^{1/2}

[(a-1-1+a^{-1})+4]^{1/2}

[(a-2+(1/a))+4]^{1/2}

[(a^{2} + 2a + 1) / a]^{1/2}

(a+1^{1/2 + 1/2}) / a^{1/2}

(a+1) / a^{1/2}

I will post #3 right away.

You can simplify to:

Hmmm, okay, can you explain how to factor in question #3. Math is not my forte, especially factoring. Sorry:)

You know that you have x as common factor.

And you have (2x-1) which are in both terms, but not in the same powers... so, take the lowest power. That's (2x-1)^-1/2

Now, this thing, I told you earlier:



It's like:



Since this is division, you can subtract the powers:



The same thing applies. Since you are factoring (2x-1)^-1/2, you are dividing by this term. Or:



Which becomes:



Is it okay? :)

Now, you can further simplify :)

Sorry, this may take a while:( Can you tell me if I am at least on the right track.

[x / (2x-1)^{-1/2}] * [x + (4x-2)]

[x^{2} + (4x^{2} - 2x)] / (2x-1)^{1/2}

This question has got me stumped:(

Um... I guess your first line was a typo from your part, which read in fact:

[x / (2x-1)^{1/2}] * [x + (4x-2)]

instead of

[x / (2x-1)^{-1/2}] * [x + (4x-2)]

This done, here is what it is:



You don't need to multiply by x now. Simplify that in the brackets:



Now, if you want, you can put it like:



or expand to give:

I have been away from my math for a few days so everything is a little fuzzy right now. I think I have a pretty good understanding of this question now. I will try a few more like it to make sure. Thank you so much for you help... and your patience!