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I am stuck on trying to solve this equation for so long and I can't seem to simplify it. I came across this step in the equation across the internet and am having a hard time trying to process how after removing the square roots we came across the new version of the equation

$$ 4 = 3(-\frac{1}{3x})^{3/2}x + 3\sqrt{-\frac{1}{3x}} + 3 $$

How did this equation above simplified into the following equation below after removing square roots,

$$ -\frac{12}{x} = \frac{64}{9x^2} - \frac{16}{3x} + 1$$

So far I have tried to isolate the square root terms on one side of the equation and then squaring both sides of the equation but this is getting me nowhere. Am I missing some basic steps?. I've also tried straight up taking square on both sides of the equation

Slime
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    Please do not rely on images. Type the formulas. https://math.stackexchange.com/help/notation – Anne Bauval May 09 '23 at 21:24
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    I advise you to set $-\frac{1}{3x}=y^2 \iff x=\frac{-1}{3y^2}$ where $y>0$. You should obtain a polynomial equation even simpler than the one which is proposed to you. – Jean Marie May 09 '23 at 21:35
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    I have removed tag "calculus" : there are neither derivatives nor integrals involved. I have also removed the tag "system of equations" because, at the beginning there is a single equation. – Jean Marie May 09 '23 at 21:36
  • Does this answer your question? Solve $\sqrt{2x-5} - \sqrt{x-1} = 1$. This is an analogous example which shows you how to "remove square roots". – Anne Bauval May 09 '23 at 21:36
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    The solution is $x=-\dfrac43$. You have first $\left(\dfrac{-1}{3x}\right)^{3/2}x^{(2/3)(3/2)}=?$ – Piquito May 09 '23 at 22:25
  • This was my first post here I am sorry if I didn't meet the context or tag guidelines I'll try to be more careful from next time onwards. – Slime May 10 '23 at 03:01
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    "tried to isolate the square root terms on one side of the equation and then squaring both sides of the equation but this is getting me nowhere" $;-;$ This will get rid of one of the square roots. Then you can repeat the steps to get rid of the remaining square root. Or, you could try the suggestion given in one of the first comments. – dxiv May 10 '23 at 03:08
  • Thanks you sm for the help I got it. Sorry for any inconveniences – Slime May 10 '23 at 03:33

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