Whakaoti i te frac{2+sqrt{3}}{2-sqrt{3}}=a+bsqrt{3}

$$\frac { 2 + \sqrt { 3 } } { 2 - \sqrt { 3 } } = a + b \sqrt { 3 }$$

Whakaoti mō b

$$b=-\frac{\sqrt{3}\left(a-4\sqrt{3}-7\right)}{3}$$

Ngā Upane mō te Whakaoti Whārite Rārangi

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}.

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

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}.

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

Pūrua 2. Pūrua \sqrt{3}.

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

Tangohia te 3 i te 4, ka 1.

$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{1}=a+b\sqrt{3}$$

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

$$\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)=a+b\sqrt{3}$$

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

$$\left(2+\sqrt{3}\right)^{2}=a+b\sqrt{3}$$

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

$$4+4\sqrt{3}+\left(\sqrt{3}\right)^{2}=a+b\sqrt{3}$$

Ko te pūrua o \sqrt{3} ko 3.

$$4+4\sqrt{3}+3=a+b\sqrt{3}$$

Tāpirihia te 4 ki te 3, ka 7.

$$7+4\sqrt{3}=a+b\sqrt{3}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a+b\sqrt{3}=7+4\sqrt{3}$$

Tangohia te a mai i ngā taha e rua.

$$b\sqrt{3}=7+4\sqrt{3}-a$$

He hanga arowhānui tō te whārite.

$$\sqrt{3}b=-a+4\sqrt{3}+7$$

Whakawehea ngā taha e rua ki te \sqrt{3}.

$$\frac{\sqrt{3}b}{\sqrt{3}}=\frac{-a+4\sqrt{3}+7}{\sqrt{3}}$$

Mā te whakawehe ki te \sqrt{3} ka wetekia te whakareanga ki te \sqrt{3}.

$$b=\frac{-a+4\sqrt{3}+7}{\sqrt{3}}$$

Whakawehe 4\sqrt{3}-a+7 ki te \sqrt{3}.

$$b=\frac{\sqrt{3}\left(-a+4\sqrt{3}+7\right)}{3}$$

Whakaoti mō a

$$a=-\sqrt{3}b+4\sqrt{3}+7$$