Whakaoti i te x^6=((6x^3)-5^3)

$${ x }^{ 6 } =((6 { x }^{ 3 } )- { 5 }^{ 3 } )$$

Whakaoti mō x (complex solution)

$$x\in \sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}},\sqrt{5}e^{-\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}},\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 5 mā te pū o 3, kia riro ko 125.

$$x^{6}=6x^{3}-125$$

Tangohia te 6x^{3} mai i ngā taha e rua.

$$x^{6}-6x^{3}=-125$$

Me tāpiri te 125 ki ngā taha e rua.

$$x^{6}-6x^{3}+125=0$$

Whakakapia te t mō te x^{3}.

$$t^{2}-6t+125=0$$

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 -6 mō te b, me te 125 mō te c i te ture pūrua.

$$t=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 1\times 125}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{6±\sqrt{-464}}{2}$$

Whakaotia te whārite t=\frac{6±\sqrt{-464}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=3+2\sqrt{29}i t=-2\sqrt{29}i+3$$

Mai i te x=t^{3}, ka taea ngā otinga mā te whakaoti te whārite mō ia t.

$$x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}} x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}} x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}} x=\sqrt{5}e^{-\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}} x=\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}} x=\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}}$$