Whakaoti i te frac{sqrt{7}+sqrt{5}}{sqrt{7}-sqrt{5}}+frac{sqrt{7}-sqrt{5}}{sqrt{7}+sqrt{5}}=

$$\frac { \sqrt { 7 } + \sqrt { 5 } } { \sqrt { 7 } - \sqrt { 5 } } + \frac { \sqrt { 7 } - \sqrt { 5 } } { \sqrt { 7 } + \sqrt { 5 } } =$$

Aromātai

$$12$$

Ngā Upane Otinga

Whakangāwaritia te tauraro o \frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{7}+\sqrt{5}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Whakaarohia te \left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\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}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Pūrua \sqrt{7}. Pūrua \sqrt{5}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{7-5}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Tangohia te 5 i te 7, ka 2.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Whakareatia te \sqrt{7}+\sqrt{5} ki te \sqrt{7}+\sqrt{5}, ka \left(\sqrt{7}+\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(\sqrt{7}+\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{7}\right)^{2}+2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Ko te pūrua o \sqrt{7} ko 7.

$$\frac{7+2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Hei whakarea \sqrt{7} me \sqrt{5}, whakareatia ngā tau i raro i te pūtake rua.

$$\frac{7+2\sqrt{35}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{7+2\sqrt{35}+5}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Tāpirihia te 7 ki te 5, ka 12.

$$\frac{12+2\sqrt{35}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Whakawehea ia wā o 12+2\sqrt{35} ki te 2, kia riro ko 6+\sqrt{35}.

$$6+\sqrt{35}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Whakangāwaritia te tauraro o \frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{7}-\sqrt{5}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}$$

Whakaarohia te \left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\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}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Pūrua \sqrt{7}. Pūrua \sqrt{5}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{7-5}$$

Tangohia te 5 i te 7, ka 2.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{2}$$

Whakareatia te \sqrt{7}-\sqrt{5} ki te \sqrt{7}-\sqrt{5}, ka \left(\sqrt{7}-\sqrt{5}\right)^{2}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)^{2}}{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{7}-\sqrt{5}\right)^{2}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}\right)^{2}-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}$$

Ko te pūrua o \sqrt{7} ko 7.

$$6+\sqrt{35}+\frac{7-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}$$

Hei whakarea \sqrt{7} me \sqrt{5}, whakareatia ngā tau i raro i te pūtake rua.

$$6+\sqrt{35}+\frac{7-2\sqrt{35}+\left(\sqrt{5}\right)^{2}}{2}$$

Ko te pūrua o \sqrt{5} ko 5.

$$6+\sqrt{35}+\frac{7-2\sqrt{35}+5}{2}$$

Tāpirihia te 7 ki te 5, ka 12.

$$6+\sqrt{35}+\frac{12-2\sqrt{35}}{2}$$

Whakawehea ia wā o 12-2\sqrt{35} ki te 2, kia riro ko 6-\sqrt{35}.

$$6+\sqrt{35}+6-\sqrt{35}$$

Tāpirihia te 6 ki te 6, ka 12.

$$12+\sqrt{35}-\sqrt{35}$$

Pahekotia te \sqrt{35} me -\sqrt{35}, ka 0.

$$12$$

Tauwehe

$$2^{2}\times 3$$