Whakaoti i te frac{4-sqrt{2}}{4+sqrt{2}}-frac{4+sqrt{2}}{4-sqrt{2}}

$$\frac { 4 - \sqrt { 2 } } { 4 + \sqrt { 2 } } - \frac { 4 + \sqrt { 2 } } { 4 - \sqrt { 2 } }$$

Aromātai

$$-\frac{8\sqrt{2}}{7}\approx -1.616244071$$

Ngā Hipanga Rongoā Poto

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

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

Whakaarohia te \left(4+\sqrt{2}\right)\left(4-\sqrt{2}\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}.

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

Pūrua 4. Pūrua \sqrt{2}.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{16-2}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Tangohia te 2 i te 16, ka 14.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Whakareatia te 4-\sqrt{2} ki te 4-\sqrt{2}, ka \left(4-\sqrt{2}\right)^{2}.

$$\frac{\left(4-\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(4-\sqrt{2}\right)^{2}.

$$\frac{16-8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Ko te pūrua o \sqrt{2} ko 2.

$$\frac{16-8\sqrt{2}+2}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Tāpirihia te 16 ki te 2, ka 18.

$$\frac{18-8\sqrt{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

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

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{\left(4-\sqrt{2}\right)\left(4+\sqrt{2}\right)}$$

Whakaarohia te \left(4-\sqrt{2}\right)\left(4+\sqrt{2}\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}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{4^{2}-\left(\sqrt{2}\right)^{2}}$$

Pūrua 4. Pūrua \sqrt{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{16-2}$$

Tangohia te 2 i te 16, ka 14.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{14}$$

Whakareatia te 4+\sqrt{2} ki te 4+\sqrt{2}, ka \left(4+\sqrt{2}\right)^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)^{2}}{14}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4+\sqrt{2}\right)^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}$$

Ko te pūrua o \sqrt{2} ko 2.

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+2}{14}$$

Tāpirihia te 16 ki te 2, ka 18.

$$\frac{18-8\sqrt{2}}{14}-\frac{18+8\sqrt{2}}{14}$$

Tā te mea he rite te tauraro o \frac{18-8\sqrt{2}}{14} me \frac{18+8\sqrt{2}}{14}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{18-8\sqrt{2}-\left(18+8\sqrt{2}\right)}{14}$$

Mahia ngā whakarea i roto o 18-8\sqrt{2}-\left(18+8\sqrt{2}\right).

$$\frac{18-8\sqrt{2}-18-8\sqrt{2}}{14}$$

Mahia ngā tātaitai i roto o 18-8\sqrt{2}-18-8\sqrt{2}.

$$\frac{-16\sqrt{2}}{14}$$

Whakawehea te -16\sqrt{2} ki te 14, kia riro ko -\frac{8}{7}\sqrt{2}.

$$-\frac{8}{7}\sqrt{2}$$