Whakaoti i te 2/sqrt{2/2}=b/frac{sqrt{2+sqrt{6}}{4}}

$$\frac { 2 } { \frac { \sqrt { 2 } } { 2 } } = \frac { b } { \frac { \sqrt { 2 } + \sqrt { 6 } } { 4 } }$$

Whakaoti mō b

$$b=\sqrt{3}+1\approx 2.732050808$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakawehe 2 ki te \frac{\sqrt{2}}{2} mā te whakarea 2 ki te tau huripoki o \frac{\sqrt{2}}{2}.

$$\frac{2\times 2}{\sqrt{2}}=\frac{b}{\frac{\sqrt{2}+\sqrt{6}}{4}}$$

Whakareatia te 2 ki te 2, ka 4.

$$\frac{4}{\sqrt{2}}=\frac{b}{\frac{\sqrt{2}+\sqrt{6}}{4}}$$

Whakangāwaritia te tauraro o \frac{4}{\sqrt{2}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{2}.

$$\frac{4\sqrt{2}}{\left(\sqrt{2}\right)^{2}}=\frac{b}{\frac{\sqrt{2}+\sqrt{6}}{4}}$$

Ko te pūrua o \sqrt{2} ko 2.

$$\frac{4\sqrt{2}}{2}=\frac{b}{\frac{\sqrt{2}+\sqrt{6}}{4}}$$

Whakawehea te 4\sqrt{2} ki te 2, kia riro ko 2\sqrt{2}.

$$2\sqrt{2}=\frac{b}{\frac{\sqrt{2}+\sqrt{6}}{4}}$$

Whakawehe b ki te \frac{\sqrt{2}+\sqrt{6}}{4} mā te whakarea b ki te tau huripoki o \frac{\sqrt{2}+\sqrt{6}}{4}.

$$2\sqrt{2}=\frac{b\times 4}{\sqrt{2}+\sqrt{6}}$$

Whakangāwaritia te tauraro o \frac{b\times 4}{\sqrt{2}+\sqrt{6}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{2}-\sqrt{6}.

$$2\sqrt{2}=\frac{b\times 4\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right)}$$

Whakaarohia te \left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$2\sqrt{2}=\frac{b\times 4\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Pūrua \sqrt{2}. Pūrua \sqrt{6}.

$$2\sqrt{2}=\frac{b\times 4\left(\sqrt{2}-\sqrt{6}\right)}{2-6}$$

Tangohia te 6 i te 2, ka -4.

$$2\sqrt{2}=\frac{b\times 4\left(\sqrt{2}-\sqrt{6}\right)}{-4}$$

Me whakakore te -4 me te -4.

$$2\sqrt{2}=b\left(-1\right)\left(\sqrt{2}-\sqrt{6}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te b\left(-1\right) ki te \sqrt{2}-\sqrt{6}.

$$2\sqrt{2}=-b\sqrt{2}+b\sqrt{6}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-b\sqrt{2}+b\sqrt{6}=2\sqrt{2}$$

Pahekotia ngā kīanga tau katoa e whai ana i te b.

$$\left(-\sqrt{2}+\sqrt{6}\right)b=2\sqrt{2}$$

He hanga arowhānui tō te whārite.

$$\left(\sqrt{6}-\sqrt{2}\right)b=2\sqrt{2}$$

Whakawehea ngā taha e rua ki te -\sqrt{2}+\sqrt{6}.

$$\frac{\left(\sqrt{6}-\sqrt{2}\right)b}{\sqrt{6}-\sqrt{2}}=\frac{2\sqrt{2}}{\sqrt{6}-\sqrt{2}}$$

Mā te whakawehe ki te -\sqrt{2}+\sqrt{6} ka wetekia te whakareanga ki te -\sqrt{2}+\sqrt{6}.

$$b=\frac{2\sqrt{2}}{\sqrt{6}-\sqrt{2}}$$

Whakawehe 2\sqrt{2} ki te -\sqrt{2}+\sqrt{6}.

$$b=\sqrt{3}+1$$