Whakaoti i te frac{9^{n}*3^{2}*3^{n}-27^n}{(3^3)^5*2^3}=1/27

$$\frac { 9 ^ { n } \times 3 ^ { 2 } \times 3 ^ { n } - 27 ^ { n } } { ( 3 ^ { 3 } ) ^ { 5 } \times 2 ^ { 3 } } = \frac { 1 } { 27 }$$

Whakaoti mō n

$$n=4$$

Ngā Upane e Whakamahi ana i te Tautuhinga o te Taupū Kōaro

Hei hiki pū ki tētahi pū anō, me whakarea ngā taupū. Me whakarea te 3 me te 5 kia riro ai te 15.

$$\frac{9^{n}\times 3^{2}\times 3^{n}-27^{n}}{3^{15}\times 2^{3}}=\frac{1}{27}$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{3^{15}\times 2^{3}}=\frac{1}{27}$$

Tātaihia te 3 mā te pū o 15, kia riro ko 14348907.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{14348907\times 2^{3}}=\frac{1}{27}$$

Tātaihia te 2 mā te pū o 3, kia riro ko 8.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{14348907\times 8}=\frac{1}{27}$$

Whakareatia te 14348907 ki te 8, ka 114791256.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256}=\frac{1}{27}$$

Tangohia te \frac{1}{27} mai i ngā taha e rua.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256}-\frac{1}{27}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 114791256 me 27 ko 114791256. Whakareatia \frac{1}{27} ki te \frac{4251528}{4251528}.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256}-\frac{4251528}{114791256}=0$$

Tā te mea he rite te tauraro o \frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256} me \frac{4251528}{114791256}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}-4251528}{114791256}=0$$

Mahia ngā whakarea i roto o 9^{n}\times 9\times 3^{n}-27^{n}-4251528.

$$\frac{9\times 27^{n}-27^{n}-4251528}{114791256}=0$$

Whakakotahitia ngā kupu rite i 9\times 27^{n}-27^{n}-4251528.

$$\frac{8\times 27^{n}-4251528}{114791256}=0$$

Whakawehea ia wā o 8\times 27^{n}-4251528 ki te 114791256, kia riro ko \frac{1}{14348907}\times 27^{n}-\frac{1}{27}.

$$\frac{1}{14348907}\times 27^{n}-\frac{1}{27}=0$$

Me tāpiri \frac{1}{27} ki ngā taha e rua o te whārite.

$$\frac{1}{14348907}\times 27^{n}=\frac{1}{27}$$

Me whakarea ngā taha e rua ki te 14348907.

$$27^{n}=531441$$

Tuhia te tau taupū kōaro o ngā taha e rua o te whārite.

$$\log(27^{n})=\log(531441)$$

Ko te taupū kōaro o tētahi tau ka hīkina ki tētahi pū ko te pū whakarea ki te taupū kōaro o taua tau.

$$n\log(27)=\log(531441)$$

Whakawehea ngā taha e rua ki te \log(27).

$$n=\frac{\log(531441)}{\log(27)}$$

Mā te tikanga tātai huri pūtake \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$n=\log_{27}\left(531441\right)$$