[Math] Rationalize and simplify $\frac{x-1}{\sqrt{2\sqrt{x}} + 1 – \sqrt[4]{x}}$

algebra-precalculusnested-radicalsradicalsrationalising-denominator

For my exercise, I have been asked to rationalize and simplify this surd;

$$\frac{x-1}{\sqrt{2\sqrt{x}} + 1 – \sqrt[4]{x}}$$

I don't know how to type it. The denominator is square root of 2 with square root of x after that is + 1 – fourth root of x.]

Each time I do this I get the wrong answer. The method I am using is;

  1133√−7×33√−733√−7

I'm confused with the $\sqrt{2}$,where $\sqrt{x}$ is inside the $\sqrt{2}$ and $1$ separated from $\sqrt{2}$ and $\sqrt{x}$. Sorry if I can't really elaborate it correctly.

This ends up nowhere near the right answer, even once it is simplified, can someone tell me where i'm going wrong?

Many thanks!

Best Answer

Note that $\sqrt{2\sqrt x}=\sqrt2\root4\of x$, so your denominator (if I have understood your comments) is $1+(\sqrt2-1)\root4\of x$. Can you rationalize now?

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