Whakaoti i te frac{sqrt{3}-1}{sqrt{3}+1}=a+bsqrt{3}

$$\frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 } = a + b \sqrt { 3 }$$

Whakaoti mō b

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

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakangāwaritia te tauraro o \frac{\sqrt{3}-1}{\sqrt{3}+1} mā te whakarea i te taurunga me te tauraro ki te \sqrt{3}-1.

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

Whakaarohia te \left(\sqrt{3}+1\right)\left(\sqrt{3}-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}.

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

Pūrua \sqrt{3}. Pūrua 1.

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

Tangohia te 1 i te 3, ka 2.

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

Whakareatia te \sqrt{3}-1 ki te \sqrt{3}-1, ka \left(\sqrt{3}-1\right)^{2}.

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

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{3}-1\right)^{2}.

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

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{3-2\sqrt{3}+1}{2}=a+b\sqrt{3}$$

Tāpirihia te 3 ki te 1, ka 4.

$$\frac{4-2\sqrt{3}}{2}=a+b\sqrt{3}$$

Whakawehea ia wā o 4-2\sqrt{3} ki te 2, kia riro ko 2-\sqrt{3}.

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

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

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

Tangohia te a mai i ngā taha e rua.

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

He hanga arowhānui tō te whārite.

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

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

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

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

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

Whakawehe -\sqrt{3}-a+2 ki te \sqrt{3}.

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

Whakaoti mō a

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