Whakaoti i te 2+1/2+1{1+frac{1/2+frac{1{A}}}}=64/27

$$2 + \frac { 1 } { 2 + \frac { 1 } { 1 + \frac { 1 } { 2 + \frac { 1 } { A } } } } = \frac { 64 } { 27 }$$

Whakaoti mō A

$$A=3$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 2 ki te \frac{A}{A}.

$$2+\frac{1}{2+\frac{1}{1+\frac{1}{\frac{2A}{A}+\frac{1}{A}}}}=\frac{64}{27}$$

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

$$2+\frac{1}{2+\frac{1}{1+\frac{1}{\frac{2A+1}{A}}}}=\frac{64}{27}$$

Tē taea kia ōrite te tāupe A ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakawehe 1 ki te \frac{2A+1}{A} mā te whakarea 1 ki te tau huripoki o \frac{2A+1}{A}.

$$2+\frac{1}{2+\frac{1}{1+\frac{A}{2A+1}}}=\frac{64}{27}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 1 ki te \frac{2A+1}{2A+1}.

$$2+\frac{1}{2+\frac{1}{\frac{2A+1}{2A+1}+\frac{A}{2A+1}}}=\frac{64}{27}$$

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

$$2+\frac{1}{2+\frac{1}{\frac{2A+1+A}{2A+1}}}=\frac{64}{27}$$

Whakakotahitia ngā kupu rite i 2A+1+A.

$$2+\frac{1}{2+\frac{1}{\frac{3A+1}{2A+1}}}=\frac{64}{27}$$

Tē taea kia ōrite te tāupe A ki -\frac{1}{2} nā te kore tautuhi i te whakawehenga mā te kore. Whakawehe 1 ki te \frac{3A+1}{2A+1} mā te whakarea 1 ki te tau huripoki o \frac{3A+1}{2A+1}.

$$2+\frac{1}{2+\frac{2A+1}{3A+1}}=\frac{64}{27}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 2 ki te \frac{3A+1}{3A+1}.

$$2+\frac{1}{\frac{2\left(3A+1\right)}{3A+1}+\frac{2A+1}{3A+1}}=\frac{64}{27}$$

Tā te mea he rite te tauraro o \frac{2\left(3A+1\right)}{3A+1} me \frac{2A+1}{3A+1}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$2+\frac{1}{\frac{2\left(3A+1\right)+2A+1}{3A+1}}=\frac{64}{27}$$

Mahia ngā whakarea i roto o 2\left(3A+1\right)+2A+1.

$$2+\frac{1}{\frac{6A+2+2A+1}{3A+1}}=\frac{64}{27}$$

Whakakotahitia ngā kupu rite i 6A+2+2A+1.

$$2+\frac{1}{\frac{8A+3}{3A+1}}=\frac{64}{27}$$

Tē taea kia ōrite te tāupe A ki -\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakawehe 1 ki te \frac{8A+3}{3A+1} mā te whakarea 1 ki te tau huripoki o \frac{8A+3}{3A+1}.

$$2+\frac{3A+1}{8A+3}=\frac{64}{27}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 2 ki te \frac{8A+3}{8A+3}.

$$\frac{2\left(8A+3\right)}{8A+3}+\frac{3A+1}{8A+3}=\frac{64}{27}$$

Tā te mea he rite te tauraro o \frac{2\left(8A+3\right)}{8A+3} me \frac{3A+1}{8A+3}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{2\left(8A+3\right)+3A+1}{8A+3}=\frac{64}{27}$$

Mahia ngā whakarea i roto o 2\left(8A+3\right)+3A+1.

$$\frac{16A+6+3A+1}{8A+3}=\frac{64}{27}$$

Whakakotahitia ngā kupu rite i 16A+6+3A+1.

$$\frac{19A+7}{8A+3}=\frac{64}{27}$$

Tē taea kia ōrite te tāupe A ki -\frac{3}{8} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 27\left(8A+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 8A+3,27.

$$27\left(19A+7\right)=64\left(8A+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 27 ki te 19A+7.

$$513A+189=64\left(8A+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 64 ki te 8A+3.

$$513A+189=512A+192$$

Tangohia te 512A mai i ngā taha e rua.

$$513A+189-512A=192$$

Pahekotia te 513A me -512A, ka A.

$$A+189=192$$

Tangohia te 189 mai i ngā taha e rua.

$$A=192-189$$

Tangohia te 189 i te 192, ka 3.

$$A=3$$