Whakaoti i te [1+1/sqrt{2}+1/2][1-1/sqrt{2}+1/2]

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

Aromātai

$$\frac{7}{4}=1.75$$

Ngā Upane Otinga

Whakangāwaritia te tauraro o \frac{1}{\sqrt{2}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{2}.

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

Ko te pūrua o \sqrt{2} ko 2.

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

Me tahuri te 1 ki te hautau \frac{2}{2}.

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

Tā te mea he rite te tauraro o \frac{2}{2} me \frac{1}{2}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

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

Tāpirihia te 2 ki te 1, ka 3.

$$\left(\frac{3}{2}+\frac{\sqrt{2}}{2}\right)\left(1-\frac{1}{\sqrt{2}}+\frac{1}{2}\right)$$

Tā te mea he rite te tauraro o \frac{3}{2} me \frac{\sqrt{2}}{2}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{3+\sqrt{2}}{2}\left(1-\frac{1}{\sqrt{2}}+\frac{1}{2}\right)$$

Whakangāwaritia te tauraro o \frac{1}{\sqrt{2}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{2}.

$$\frac{3+\sqrt{2}}{2}\left(1-\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\frac{1}{2}\right)$$

Ko te pūrua o \sqrt{2} ko 2.

$$\frac{3+\sqrt{2}}{2}\left(1-\frac{\sqrt{2}}{2}+\frac{1}{2}\right)$$

Me tahuri te 1 ki te hautau \frac{2}{2}.

$$\frac{3+\sqrt{2}}{2}\left(\frac{2}{2}-\frac{\sqrt{2}}{2}+\frac{1}{2}\right)$$

Tā te mea he rite te tauraro o \frac{2}{2} me \frac{1}{2}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{3+\sqrt{2}}{2}\left(\frac{2+1}{2}-\frac{\sqrt{2}}{2}\right)$$

Tāpirihia te 2 ki te 1, ka 3.

$$\frac{3+\sqrt{2}}{2}\left(\frac{3}{2}-\frac{\sqrt{2}}{2}\right)$$

Tā te mea he rite te tauraro o \frac{3}{2} me \frac{\sqrt{2}}{2}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{3+\sqrt{2}}{2}\times \frac{3+\sqrt{2}}{2}$$

Whakareatia te \frac{3+\sqrt{2}}{2} ki te \frac{3+\sqrt{2}}{2}, ka \left(\frac{3+\sqrt{2}}{2}\right)^{2}.

$$\left(\frac{3+\sqrt{2}}{2}\right)^{2}$$

Kia whakarewa i te \frac{3+\sqrt{2}}{2} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$\frac{\left(3+\sqrt{2}\right)^{2}}{2^{2}}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3+\sqrt{2}\right)^{2}.

$$\frac{9+6\sqrt{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Ko te pūrua o \sqrt{2} ko 2.

$$\frac{9+6\sqrt{2}+2}{2^{2}}$$

Tāpirihia te 9 ki te 2, ka 11.

$$\frac{11+6\sqrt{2}}{2^{2}}$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$\frac{11+6\sqrt{2}}{4}$$

Tauwehe

$$\frac{7}{2 ^ {2}} = 1\frac{3}{4} = 1.75$$