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

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

Aromātai

$$\frac{32}{9}\approx 3.555555556$$

Ngā Hipanga Rongoā Poto

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

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

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

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

Pūrua 5. Pūrua \sqrt{7}.

$$\frac{\left(5-\sqrt{7}\right)\left(5-\sqrt{7}\right)}{25-7}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Tangohia te 7 i te 25, ka 18.

$$\frac{\left(5-\sqrt{7}\right)\left(5-\sqrt{7}\right)}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

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

$$\frac{\left(5-\sqrt{7}\right)^{2}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

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

$$\frac{25-10\sqrt{7}+\left(\sqrt{7}\right)^{2}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Ko te pūrua o \sqrt{7} ko 7.

$$\frac{25-10\sqrt{7}+7}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Tāpirihia te 25 ki te 7, ka 32.

$$\frac{32-10\sqrt{7}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{\left(5-\sqrt{7}\right)\left(5+\sqrt{7}\right)}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{5^{2}-\left(\sqrt{7}\right)^{2}}$$

Pūrua 5. Pūrua \sqrt{7}.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{25-7}$$

Tangohia te 7 i te 25, ka 18.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{18}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)^{2}}{18}$$

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

$$\frac{32-10\sqrt{7}}{18}+\frac{25+10\sqrt{7}+\left(\sqrt{7}\right)^{2}}{18}$$

Ko te pūrua o \sqrt{7} ko 7.

$$\frac{32-10\sqrt{7}}{18}+\frac{25+10\sqrt{7}+7}{18}$$

Tāpirihia te 25 ki te 7, ka 32.

$$\frac{32-10\sqrt{7}}{18}+\frac{32+10\sqrt{7}}{18}$$

Tā te mea he rite te tauraro o \frac{32-10\sqrt{7}}{18} me \frac{32+10\sqrt{7}}{18}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{32-10\sqrt{7}+32+10\sqrt{7}}{18}$$

Mahia ngā tātaitai i roto o 32-10\sqrt{7}+32+10\sqrt{7}.

$$\frac{64}{18}$$

Whakahekea te hautanga \frac{64}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$\frac{32}{9}$$

Tauwehe

$$\frac{2 ^ {5}}{3 ^ {2}} = 3\frac{5}{9} = 3.5555555555555554$$