Whakaoti i te 2/sqrt{2-2}+frac{sqrt{2}+1}{sqrt{2}-1}-frac{sqrt{32}}{2}

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

Aromātai

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

Ngā Upane Otinga

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

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

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

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

Pūrua \sqrt{2}. Pūrua 2.

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

Tangohia te 4 i te 2, ka -2.

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

Me whakakore te -2 me te -2.

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

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

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

Whakaarohia te \left(\sqrt{2}-1\right)\left(\sqrt{2}+1\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}.

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

Pūrua \sqrt{2}. Pūrua 1.

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

Tangohia te 1 i te 2, ka 1.

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

Ka whakawehea he tau ki te tahi, hua ai ko ia anō.

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

Whakareatia te \sqrt{2}+1 ki te \sqrt{2}+1, ka \left(\sqrt{2}+1\right)^{2}.

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

Tauwehea te 32=4^{2}\times 2. Tuhia anō te pūtake rua o te hua \sqrt{4^{2}\times 2} hei hua o ngā pūtake rua \sqrt{4^{2}}\sqrt{2}. Tuhia te pūtakerua o te 4^{2}.

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

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

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

Hei kimi i te tauaro o \sqrt{2}+2, kimihia te tauaro o ia taurangi.

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(\sqrt{2}+1\right)^{2}.

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

Ko te pūrua o \sqrt{2} ko 2.

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

Tāpirihia te 2 ki te 1, ka 3.

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

Tāpirihia te -2 ki te 3, ka 1.

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

Pahekotia te -\sqrt{2} me 2\sqrt{2}, ka \sqrt{2}.

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

Pahekotia te \sqrt{2} me -2\sqrt{2}, ka -\sqrt{2}.

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

Tauwehe

$$1-\sqrt{2}$$