Whakaoti i te 3y^3=81

$$3 y ^ { 3 } = 81$$

Whakaoti mō y (complex solution)

$$y=\frac{-3\sqrt{3}i-3}{2}\approx -1.5-2.598076211i$$

$$y=3$$

$$y=\frac{-3+3\sqrt{3}i}{2}\approx -1.5+2.598076211i$$

Ngā Upane Otinga

Whakawehea ngā taha e rua ki te 3.

$$y^{3}=\frac{81}{3}$$

Whakawehea te 81 ki te 3, kia riro ko 27.

$$y^{3}=27$$

Tangohia te 27 mai i ngā taha e rua.

$$y^{3}-27=0$$

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau -27, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±27,±9,±3,±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$y=3$$

Mā te whakatakotoranga Tauwehe, he tauwehe te y-k o te pūrau mō ia pūtake k. Whakawehea te y^{3}-27 ki te y-3, kia riro ko y^{2}+3y+9. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$y^{2}+3y+9=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 1 mō te a, te 3 mō te b, me te 9 mō te c i te ture pūrua.

$$y=\frac{-3±\sqrt{3^{2}-4\times 1\times 9}}{2}$$

Mahia ngā tātaitai.

$$y=\frac{-3±\sqrt{-27}}{2}$$

Whakaotia te whārite y^{2}+3y+9=0 ina he tōrunga te ±, ina he tōraro te ±.

$$y=\frac{-3i\sqrt{3}-3}{2} y=\frac{-3+3i\sqrt{3}}{2}$$

Rārangitia ngā otinga katoa i kitea.

$$y=3 y=\frac{-3i\sqrt{3}-3}{2} y=\frac{-3+3i\sqrt{3}}{2}$$

Whakaoti mō y

$$y=3$$

Ngā Upane Otinga

Whakawehea ngā taha e rua ki te 3.

$$y^{3}=\frac{81}{3}$$

Whakawehea te 81 ki te 3, kia riro ko 27.

$$y^{3}=27$$

Tangohia te 27 mai i ngā taha e rua.

$$y^{3}-27=0$$

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau -27, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±27,±9,±3,±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$y=3$$

Mā te whakatakotoranga Tauwehe, he tauwehe te y-k o te pūrau mō ia pūtake k. Whakawehea te y^{3}-27 ki te y-3, kia riro ko y^{2}+3y+9. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$y^{2}+3y+9=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 1 mō te a, te 3 mō te b, me te 9 mō te c i te ture pūrua.

$$y=\frac{-3±\sqrt{3^{2}-4\times 1\times 9}}{2}$$

Mahia ngā tātaitai.

$$y=\frac{-3±\sqrt{-27}}{2}$$

Tā te mea e kore te pūrua o tētahi tau tōraro e tautohutia ki te āpure tūturu, kāhore he rongoā.

$$y\in \emptyset$$

Rārangitia ngā otinga katoa i kitea.

$$y=3$$