Understanding Fraction Simplification

[shamieh]

Confused on how we go from

\(\displaystyle \frac{1}{^4\sqrt{1 + x}}\) to \(\displaystyle \frac{4}{3}(1 + x)^\frac{3}{4}\)

Can someone please show me step-by-step. I need to see the basic steps.

Thanks in advance.

 

[Prove It]

Confused on how we go from

\(\displaystyle \frac{1}{^4\sqrt{1 + x}}\) to \(\displaystyle \frac{4}{3}(1 + x)^\frac{3}{4}\)

Can someone please show me step-by-step. I need to see the basic steps.

Thanks in advance.

These things are NOT the same, so you can't "convert" them...

$\displaystyle \begin{align*} \frac{1}{\sqrt[4]{1 + x}} &= \frac{1}{ \left( 1 + x \right) ^{\frac{1}{4}} } \\ &= \left( 1 + x \right) ^{-\frac{1}{4}} \end{align*}$

It APPEARS though that you are trying to ANTIDIFFERENTIATE (Integrate) this function, which you should be able to do now...

 

[shamieh]

Ahh! Thank you!(Yes)

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And this particular problem would be divergent since you would get \(\displaystyle a^{3/4}\)which is > 1 correct?

 

[Prove It]

Ahh! Thank you!(Yes)

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And this particular problem would be divergent since you would get \(\displaystyle a^{3/4}\)which is > 1 correct?

What on EARTH are you talking about? WHAT is divergent? WHAT are you actually trying to do with this question?

 

[shamieh]

Oh sorry the original problem is the equation up top as \(\displaystyle \int^\infty_0\) and it's improper so i rewrote it as \(\displaystyle \int^a_0\) thus; \(\displaystyle \lim_{a\to\infty}\) and I ended up with a underneath the \(\displaystyle \sqrt{} \)to the \(\displaystyle ^3\) power.

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The initial question of the problem was Determine whether or not the integral is convergent or divergent. Sorry, forgot to include that.

So essentially I had this \(\displaystyle \lim_{a\to\infty} \frac{4}{3}(1 + a)^{3/4} - \frac{4}{3}\) so I'm guessing since it's \(\displaystyle \infty\) in the square root it's always going to keep growing no matter what and be Divergent