Datrys 2/sqrt{2-2}+frac{sqrt{2}+1}{sqrt{2}-1}-frac{sqrt{32}}{2}

$$\frac { 2 } { \sqrt { 2 } - 2 } + \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 } - \frac { \sqrt { 32 } } { 2 }$$

Enrhifo

$$1-\sqrt{2}\approx -0.414213562$$

Camau Ateb

Mae'n rhesymoli enwadur \frac{2}{\sqrt{2}-2} drwy luosi'r rhifiadur a'r enwadur â \sqrt{2}+2.

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}-2\right)\left(\sqrt{2}+2\right)}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Ystyriwch \left(\sqrt{2}-2\right)\left(\sqrt{2}+2\right). Gellir trawsnewid lluosi yn wahaniaeth rhwng sgwariau drwy ddefnyddio’r rheol: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Sgwâr \sqrt{2}. Sgwâr 2.

$$\frac{2\left(\sqrt{2}+2\right)}{2-4}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Tynnu 4 o 2 i gael -2.

$$\frac{2\left(\sqrt{2}+2\right)}{-2}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Canslo -2 a -2.

$$-\left(\sqrt{2}+2\right)+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Mae'n rhesymoli enwadur \frac{\sqrt{2}+1}{\sqrt{2}-1} drwy luosi'r rhifiadur a'r enwadur â \sqrt{2}+1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}-\frac{\sqrt{32}}{2}$$

Ystyriwch \left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right). Gellir trawsnewid lluosi yn wahaniaeth rhwng sgwariau drwy ddefnyddio’r rheol: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}-\frac{\sqrt{32}}{2}$$

Sgwâr \sqrt{2}. Sgwâr 1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{2-1}-\frac{\sqrt{32}}{2}$$

Tynnu 1 o 2 i gael 1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{1}-\frac{\sqrt{32}}{2}$$

Mae rhannu unrhyw beth ag un yn rhoi'r rhif hwnnw.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)-\frac{\sqrt{32}}{2}$$

Lluosi \sqrt{2}+1 a \sqrt{2}+1 i gael \left(\sqrt{2}+1\right)^{2}.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)^{2}-\frac{\sqrt{32}}{2}$$

Ffactora 32=4^{2}\times 2. Ailysgrifennu ail isradd y lluoswm \sqrt{4^{2}\times 2} fel lluoswm ail israddau \sqrt{4^{2}}\sqrt{2}. Cymryd isradd 4^{2}.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)^{2}-\frac{4\sqrt{2}}{2}$$

Rhannu 4\sqrt{2} â 2 i gael 2\sqrt{2}.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)^{2}-2\sqrt{2}$$

I ddod o hyd i wrthwyneb \sqrt{2}+2, dewch o hyd i wrthwyneb pob term.

$$-\sqrt{2}-2+\left(\sqrt{2}+1\right)^{2}-2\sqrt{2}$$

Defnyddio'r theorem binomaidd \left(a+b\right)^{2}=a^{2}+2ab+b^{2} i ehangu'r \left(\sqrt{2}+1\right)^{2}.

$$-\sqrt{2}-2+\left(\sqrt{2}\right)^{2}+2\sqrt{2}+1-2\sqrt{2}$$

Sgwâr \sqrt{2} yw 2.

$$-\sqrt{2}-2+2+2\sqrt{2}+1-2\sqrt{2}$$

Adio 2 a 1 i gael 3.

$$-\sqrt{2}-2+3+2\sqrt{2}-2\sqrt{2}$$

Adio -2 a 3 i gael 1.

$$-\sqrt{2}+1+2\sqrt{2}-2\sqrt{2}$$

Cyfuno -\sqrt{2} a 2\sqrt{2} i gael \sqrt{2}.

$$\sqrt{2}+1-2\sqrt{2}$$

Cyfuno \sqrt{2} a -2\sqrt{2} i gael -\sqrt{2}.

$$-\sqrt{2}+1$$

Ffactor

$$1-\sqrt{2}$$