Whakaoti i te (sqrt{18}+sqrt{12})(3sqrt{2}-2sqrt{3})-(sqrt{3}-sqrt{2})^2

$$( \sqrt { 18 } + \sqrt { 12 } ) ( 3 \sqrt { 2 } - 2 \sqrt { 3 } ) - ( \sqrt { 3 } - \sqrt { 2 } ) ^ { 2 }$$

Aromātai

$$2\sqrt{6}+1\approx 5.898979486$$

Ngā Upane Otinga

Tauwehea te 18=3^{2}\times 2. Tuhia anō te pūtake rua o te hua \sqrt{3^{2}\times 2} hei hua o ngā pūtake rua \sqrt{3^{2}}\sqrt{2}. Tuhia te pūtakerua o te 3^{2}.

$$\left(3\sqrt{2}+\sqrt{12}\right)\left(3\sqrt{2}-2\sqrt{3}\right)-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Tauwehea te 12=2^{2}\times 3. Tuhia anō te pūtake rua o te hua \sqrt{2^{2}\times 3} hei hua o ngā pūtake rua \sqrt{2^{2}}\sqrt{3}. Tuhia te pūtakerua o te 2^{2}.

$$\left(3\sqrt{2}+2\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Whakaarohia te \left(3\sqrt{2}+2\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\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}.

$$\left(3\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Whakarohaina te \left(3\sqrt{2}\right)^{2}.

$$3^{2}\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

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

$$9\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Ko te pūrua o \sqrt{2} ko 2.

$$9\times 2-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Whakareatia te 9 ki te 2, ka 18.

$$18-\left(2\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Whakarohaina te \left(2\sqrt{3}\right)^{2}.

$$18-2^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$18-4\left(\sqrt{3}\right)^{2}-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Ko te pūrua o \sqrt{3} ko 3.

$$18-4\times 3-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Whakareatia te 4 ki te 3, ka 12.

$$18-12-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Tangohia te 12 i te 18, ka 6.

$$6-\left(\sqrt{3}-\sqrt{2}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{3}-\sqrt{2}\right)^{2}.

$$6-\left(\left(\sqrt{3}\right)^{2}-2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

Ko te pūrua o \sqrt{3} ko 3.

$$6-\left(3-2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}\right)$$

Hei whakarea \sqrt{3} me \sqrt{2}, whakareatia ngā tau i raro i te pūtake rua.

$$6-\left(3-2\sqrt{6}+\left(\sqrt{2}\right)^{2}\right)$$

Ko te pūrua o \sqrt{2} ko 2.

$$6-\left(3-2\sqrt{6}+2\right)$$

Tāpirihia te 3 ki te 2, ka 5.

$$6-\left(5-2\sqrt{6}\right)$$

Hei kimi i te tauaro o 5-2\sqrt{6}, kimihia te tauaro o ia taurangi.

$$6-5+2\sqrt{6}$$

Tangohia te 5 i te 6, ka 1.

$$1+2\sqrt{6}$$