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Mathematics 54 Online
OpenStudy (konradzuse):

Question about "Expanding the integral."

OpenStudy (konradzuse):

so I'm curious how we get things like 1/(sqrt(x^4) = sqrt(x^4)/x^4

OpenStudy (konradzuse):

or that sqrt(x^4)/x^4 = 1/x^2

OpenStudy (konradzuse):

Is this one approach to many for doing a question like this? Sometimes Wolfram does some "interesting" things...

OpenStudy (anonymous):

i dont know about the integral part, but sqrt(x^4)/x^4 = 1/x^2 because the square root of x^4 is x^2. and then the bottom can become x^2. so its 1/x^2

OpenStudy (konradzuse):

Ty Jessica.

OpenStudy (lgbasallote):

are you familiar with the magic \[\frac{1}{\sqrt x} \implies \frac{1}{\sqrt x} \times \frac{\sqrt x}{\sqrt x} \implies \frac{\sqrt x}{x}\]

OpenStudy (lgbasallote):

it follows the same principle

OpenStudy (konradzuse):

no.

OpenStudy (lgbasallote):

you're not familiar with rationalization?

OpenStudy (konradzuse):

teach me about everything.

OpenStudy (lgbasallote):

i find that very surprising...

OpenStudy (konradzuse):

hmm now that I look at that I think I've seen that before... So we do this with sqrt's usually?

OpenStudy (lgbasallote):

rationalization is done when you have a radical in the denominator..your goal is to "neutralize" the denominator and depends on the situation really

OpenStudy (konradzuse):

hmm cool trick :)

OpenStudy (konradzuse):

I have been denied this cool tricks my entire childhood.

OpenStudy (lgbasallote):

for example you have \[\int \frac{x-1}{\sqrt x}dx\] there's no need for rationalization...just split up the integral \[\int \frac{x}{\sqrt x} - \int \frac{1}{\sqrt x}\] you can use rationalization but you know this is just \[\int x^{1 - 1/2} - \int x^{-1/2}\]

OpenStudy (lgbasallote):

\[x^{1 - 1/2} = x^{1/2}\] now notice if i use rationalization \[\frac{x}{\sqrt x} \implies \frac{x}{\sqrt x} \times \frac{\sqrt x}{\sqrt x} \implies \frac{x\sqrt x}{x} \implies \sqrt x \implies x^{1/2}\]

OpenStudy (lgbasallote):

both will result the same answers

OpenStudy (konradzuse):

so if you had a x^1/2 you could randomly do x/(x)^(1/2)

OpenStudy (konradzuse):

you're soo cool iggy.

OpenStudy (konradzuse):

TEACH ME MOAR.

OpenStudy (konradzuse):

@lgbasallote run away after medal eh?

OpenStudy (lgbasallote):

lol i ran away much earlier :p

OpenStudy (lgbasallote):

and why would you make x^1/2 into a more complicated state?

OpenStudy (lgbasallote):

i guess if you have some outside of the box plan...it's possible

sam (.sam.):

I'm getting this from another calculator

sam (.sam.):

I'm sure there's something to do with "restricted values of x" in wolfram

sam (.sam.):

wolfram

OpenStudy (konradzuse):

:)

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