Thread: Radicals and Exponent Properties

Hello all, wondering if you could help me. I am brushing up on math to go back to school, and just looked over a practice test for algebra. The only question I couldn't figure out how to answer was this one:

[Sorry about the formatting. The whole fraction is under the radical here, in case this is ambiguous. I'm willing to make anything look neater if it's explained how.]

For all x>0 and y /=/ 0, which of the following expressions is equivalent to √x^4/ y^5 ?


> A) x^2 √y/y^3

> B) x^2 √y/y^3

> C) x^2/y^3

> D) x^2/ ³√y

> E) x^2 √y


I looked up the answer, and it still doesn't really make sense to me. Like, I know some basics about exponents. Like I think the square root of the exponent is the exponent times 1/2, which would give me x^2/y^(2.5), right? But I can't figure out how to relate that to these options, and clearly have some big gaps in my knowledge here. Any help would be appreciated.


(Again, sorry about the formatting. I know it's hard for me to read that)

Originally Posted by SkittlesBFresh

Like I think the square root of the exponent is the exponent times 1/2,
which would give me x^2/y^(2.5), right?

Agree.

Originally Posted by highvoltage

OK, I feel like I am almost getting it, but in step 4, I'm not understanding how we got that √y in the numerator.

(Thanks for the help so far, btw. I really appreciate that a stranger on the internet took the time to do this.)

Edit: Oh, wait, I see, just multiplied both by √y? I'm not sure if I'm confused or not, lol.

Edit II: OK, now I got it. I sort of got it with edit I, but I just reconciled it with my statement that Wilmer confirmed, which made it click more somehow. I don't know what I'm talking about, but I get this answer now. Thanks.

Skittles, what was done by multiplying the numerator and denominator by SQRT(y)
is really a net multiplication by 1 : SQRT(y) / SQRT(y) = 1

This is a process known as "rationalizing the denominator": in other words, if there
is a SQRT(something) in denominator, get it out of there!

If you really want to see the why/how, go here:

OK, that makes even more sense then. My assumption was that it was just to because they didn't want a weird exponent (2 1/2), or else it was just sort of done to do it, and make you figure out which was the same.

I do remember this now, thank you.

(What is the end game for doing this? Like, I didn't get why we learned to factor >1st degree polynomials until I got to the problems where I needed it. Just sort of in general terms, when does it help up to have the radical in the numerator instead of the denominator?)

And thanks again for the help.

Originally Posted by SkittlesBFresh

(What is the end game for doing this? Like, I didn't get why we learned to factor >1st degree polynomials until I got to the problems where I needed it. Just sort of in general terms, when does it help up to have the radical in the numerator instead of the denominator?)

Dunno. Looks better?

lol, fair enough.