Whakaoti i te (sqrt{5}-sqrt{3})(sqrt{5}+sqrt{3})-(sqrt{6}+sqrt{2})^2

$$( \sqrt { 5 } - \sqrt { 3 } ) ( \sqrt { 5 } + \sqrt { 3 } ) - ( \sqrt { 6 } + \sqrt { 2 } ) ^ { 2 }$$

Aromātai

$$-4\sqrt{3}-6\approx -12.92820323$$

Ngā Upane Otinga

Whakaarohia te \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\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}.

$$\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}-\left(\sqrt{6}+\sqrt{2}\right)^{2}$$

Ko te pūrua o \sqrt{5} ko 5.

$$5-\left(\sqrt{3}\right)^{2}-\left(\sqrt{6}+\sqrt{2}\right)^{2}$$

Ko te pūrua o \sqrt{3} ko 3.

$$5-3-\left(\sqrt{6}+\sqrt{2}\right)^{2}$$

Tangohia te 3 i te 5, ka 2.

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(\sqrt{6}+\sqrt{2}\right)^{2}.

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

Ko te pūrua o \sqrt{6} ko 6.

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

Tauwehea te 6=2\times 3. Tuhia anō te pūtake rua o te hua \sqrt{2\times 3} hei hua o ngā pūtake rua \sqrt{2}\sqrt{3}.

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

Whakareatia te \sqrt{2} ki te \sqrt{2}, ka 2.

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

Whakareatia te 2 ki te 2, ka 4.

$$2-\left(6+4\sqrt{3}+\left(\sqrt{2}\right)^{2}\right)$$

Ko te pūrua o \sqrt{2} ko 2.

$$2-\left(6+4\sqrt{3}+2\right)$$

Tāpirihia te 6 ki te 2, ka 8.

$$2-\left(8+4\sqrt{3}\right)$$

Hei kimi i te tauaro o 8+4\sqrt{3}, kimihia te tauaro o ia taurangi.

$$2-8-4\sqrt{3}$$

Tangohia te 8 i te 2, ka -6.

$$-6-4\sqrt{3}$$