The solution to the division of the given surd is: [tex]\mathbf{P =\dfrac{(6\sqrt{x}-x^2\sqrt{x}-3x\sqrt{x}+2x+2)(x-1) }{8x} }[/tex]
The division of surds follows a systemic approach whereby we divide the whole numbers separately and the root(s) are being divided by each other.
Given that:
[tex]\mathbf{P=(\frac{\sqrt{x} +2}{x-1}+\frac{\sqrt{x}\\ -2}{x-2\sqrt{x} +1} ) : \frac{4x}{(x-1)^{2} }}[/tex]
i.e.
[tex]\mathbf{=\dfrac{(\frac{\sqrt{x} +2}{x-1}+\frac{\sqrt{x}\\ -2}{x-2\sqrt{x} +1} )}{ \frac{4x}{(x-1)^{2} }} }[/tex]
Using the fraction rule:
[tex]\mathbf{\dfrac{a}{\dfrac{b}{c}}= \dfrac{a\times c}{b}}[/tex]
[tex]\mathbf{\implies \dfrac{(\frac{\sqrt{x} +2}{x-1}+\frac{\sqrt{x}\\ -2}{x-2\sqrt{x} +1} )(x-1)^{2}}{4x}} }[/tex]
By simplification, we have:
[tex]\mathbf{ =\dfrac{\dfrac{(6\sqrt{x}-x^2\sqrt{x}-3x\sqrt{x}+2x+2)(x-1) }{2} }{4x} }[/tex]
[tex]\mathbf{P =\dfrac{(6\sqrt{x}-x^2\sqrt{x}-3x\sqrt{x}+2x+2)(x-1) }{8x} }[/tex]
Learn more about evaluating the division of surds here:
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