forkorte brøk

[Guest]

Hvordan forkorter man denne stygge brøken [tex]\frac{x^{1/2}-x^{-1/2}}{x^{1/2}+x^{-1/2}}[/tex] ?
takker for svar.

[Drezky]

[tex]\frac{x^{1/2}-x^{-1/2}}{x^{1/2}+x^{-1/2}}=\frac{\sqrt{x}-\frac{1}{\sqrt{x}}}{\sqrt{x}+\frac{1}{\sqrt{x}}}=\frac{\frac{x-1}{\sqrt{x}}}{\frac{x+1}{\sqrt{x}}}=\frac{x-1}{x+1}[/tex]

Voilà

[Guest]

takk. men kan du forklare litt stegene? (ta med mellomregning)

[Drezky]

[tex]\frac{\sqrt{x}-\frac{1}{\sqrt{x}}}{\sqrt{x}+\frac{1}{\sqrt{x}}}=\frac{teller}{nevner}[/tex]
[tex]\sqrt{x}-\frac{1}{\sqrt{x}}=\frac{\sqrt{x}}{1}-\frac{1}{\sqrt{x}}[/tex]
[tex]teller:\:\sqrt{x}-\frac{1}{\sqrt{x}}:mfm\left (\frac{\sqrt{x}}{1}-\frac{1}{\sqrt{x}} \right )=\sqrt{x}[/tex]

[tex]\frac{\sqrt{x}*\sqrt{x}}{\sqrt{x}}-\frac{1}{\sqrt{x}}=\frac{\left ( \sqrt{x} \right )^2-1}{\sqrt{x}}=\frac{x-1}{\sqrt{x}}[/tex]

Gjør samme med nevner...

[tex]\frac{\frac{x-1}{\sqrt{x}}}{\frac{x+1}{\sqrt{x}}}=\frac{x-1}{\sqrt{x}}*\frac{\sqrt{x}}{x+1}=\frac{x-1}{x+1}[/tex]

Alternativt kan du bare multiplisere med [tex]\sqrt{x}[/tex] fra starten av:

[tex]\frac{\left ( \sqrt{x}-\frac{1}{\sqrt{x}} \right )*{\color{Blue} {\sqrt{x}}}}{\left ( \sqrt{x}+\frac{1}{\sqrt{x}} \right )*{\color{Blue} {\sqrt{x}}}}=\frac{x-1}{x+1}[/tex]