Whakaoti i te (sqrt{2}+sqrt{5})^2-(2+sqrt{10})^2+sqrt{90}+(2sqrt{2}-1)(2sqrt{2}+1)

$$( \sqrt { 2 } + \sqrt { 5 } ) ^ { 2 } - ( 2 + \sqrt { 10 } ) ^ { 2 } + \sqrt { 90 } + ( 2 \sqrt { 2 } - 1 ) ( 2 \sqrt { 2 } + 1 )$$

Aromātai

$$\sqrt{10}\approx 3.16227766$$

Ngā Upane Otinga

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(\sqrt{2}+\sqrt{5}\right)^{2}.

$$\left(\sqrt{2}\right)^{2}+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Ko te pūrua o \sqrt{2} ko 2.

$$2+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Hei whakarea \sqrt{2} me \sqrt{5}, whakareatia ngā tau i raro i te pūtake rua.

$$2+2\sqrt{10}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Ko te pūrua o \sqrt{5} ko 5.

$$2+2\sqrt{10}+5-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Tāpirihia te 2 ki te 5, ka 7.

$$7+2\sqrt{10}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2+\sqrt{10}\right)^{2}.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+\left(\sqrt{10}\right)^{2}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Ko te pūrua o \sqrt{10} ko 10.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+10\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Tāpirihia te 4 ki te 10, ka 14.

$$7+2\sqrt{10}-\left(14+4\sqrt{10}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Hei kimi i te tauaro o 14+4\sqrt{10}, kimihia te tauaro o ia taurangi.

$$7+2\sqrt{10}-14-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Tangohia te 14 i te 7, ka -7.

$$-7+2\sqrt{10}-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Pahekotia te 2\sqrt{10} me -4\sqrt{10}, ka -2\sqrt{10}.

$$-7-2\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Tauwehea te 90=3^{2}\times 10. Tuhia anō te pūtake rua o te hua \sqrt{3^{2}\times 10} hei hua o ngā pūtake rua \sqrt{3^{2}}\sqrt{10}. Tuhia te pūtakerua o te 3^{2}.

$$-7-2\sqrt{10}+3\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Pahekotia te -2\sqrt{10} me 3\sqrt{10}, ka \sqrt{10}.

$$-7+\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Whakaarohia te \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\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}. Pūrua 1.

$$-7+\sqrt{10}+\left(2\sqrt{2}\right)^{2}-1$$

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

$$-7+\sqrt{10}+2^{2}\left(\sqrt{2}\right)^{2}-1$$

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

$$-7+\sqrt{10}+4\left(\sqrt{2}\right)^{2}-1$$

Ko te pūrua o \sqrt{2} ko 2.

$$-7+\sqrt{10}+4\times 2-1$$

Whakareatia te 4 ki te 2, ka 8.

$$-7+\sqrt{10}+8-1$$

Tangohia te 1 i te 8, ka 7.

$$-7+\sqrt{10}+7$$

Tāpirihia te -7 ki te 7, ka 0.

$$\sqrt{10}$$