Whakaoti i te frac{2+sqrt{3}}{2-sqrt{3}}-frac{2-sqrt{3}}{2+sqrt{3}}

$$\frac{ 2+ \sqrt{ 3 } }{ 2- \sqrt{ 3 } } - \frac{ 2- \sqrt{ 3 } }{ 2+ \sqrt{ 3 } }$$

Aromātai

$$8\sqrt{3}\approx 13.856406461$$

Ngā Upane Otinga

Whakangāwaritia te tauraro o \frac{2+\sqrt{3}}{2-\sqrt{3}} mā te whakarea i te taurunga me te tauraro ki te 2+\sqrt{3}.

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

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

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

Pūrua 2. Pūrua \sqrt{3}.

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

Tangohia te 3 i te 4, ka 1.

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

Ka whakawehea he tau ki te tahi, hua ai ko ia anō.

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

Whakareatia te 2+\sqrt{3} ki te 2+\sqrt{3}, ka \left(2+\sqrt{3}\right)^{2}.

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

Whakangāwaritia te tauraro o \frac{2-\sqrt{3}}{2+\sqrt{3}} mā te whakarea i te taurunga me te tauraro ki te 2-\sqrt{3}.

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

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

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

Pūrua 2. Pūrua \sqrt{3}.

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

Tangohia te 3 i te 4, ka 1.

$$\left(2+\sqrt{3}\right)^{2}-\frac{\left(2-\sqrt{3}\right)\left(2-\sqrt{3}\right)}{1}$$

Ka whakawehea he tau ki te tahi, hua ai ko ia anō.

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

Whakareatia te 2-\sqrt{3} ki te 2-\sqrt{3}, ka \left(2-\sqrt{3}\right)^{2}.

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

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

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

Ko te pūrua o \sqrt{3} ko 3.

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

Tāpirihia te 4 ki te 3, ka 7.

$$7+4\sqrt{3}-\left(2-\sqrt{3}\right)^{2}$$

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

$$7+4\sqrt{3}-\left(4-4\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)$$

Ko te pūrua o \sqrt{3} ko 3.

$$7+4\sqrt{3}-\left(4-4\sqrt{3}+3\right)$$

Tāpirihia te 4 ki te 3, ka 7.

$$7+4\sqrt{3}-\left(7-4\sqrt{3}\right)$$

Hei kimi i te tauaro o 7-4\sqrt{3}, kimihia te tauaro o ia taurangi.

$$7+4\sqrt{3}-7-\left(-4\sqrt{3}\right)$$

Ko te tauaro o -4\sqrt{3} ko 4\sqrt{3}.

$$7+4\sqrt{3}-7+4\sqrt{3}$$

Tangohia te 7 i te 7, ka 0.

$$4\sqrt{3}+4\sqrt{3}$$

Pahekotia te 4\sqrt{3} me 4\sqrt{3}, ka 8\sqrt{3}.

$$8\sqrt{3}$$