Whakaoti i te frac{5+2sqrt{3}}{7+4sqrt{3}}=x+sqrt{3}y

$$\frac { 5 + 2 \sqrt { 3 } } { 7 + 4 \sqrt { 3 } } = x + \sqrt { 3 } y$$

Whakaoti mō y

$$y=-\frac{\sqrt{3}\left(x+6\sqrt{3}-11\right)}{3}$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakangāwaritia te tauraro o \frac{5+2\sqrt{3}}{7+4\sqrt{3}} mā te whakarea i te taurunga me te tauraro ki te 7-4\sqrt{3}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}=x+\sqrt{3}y$$

Whakaarohia te \left(7+4\sqrt{3}\right)\left(7-4\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}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{7^{2}-\left(4\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Tātaihia te 7 mā te pū o 2, kia riro ko 49.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-\left(4\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Whakarohaina te \left(4\sqrt{3}\right)^{2}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-4^{2}\left(\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}=x+\sqrt{3}y$$

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\times 3}=x+\sqrt{3}y$$

Whakareatia te 16 ki te 3, ka 48.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-48}=x+\sqrt{3}y$$

Tangohia te 48 i te 49, ka 1.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{1}=x+\sqrt{3}y$$

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

$$\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)=x+\sqrt{3}y$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5+2\sqrt{3} ki te 7-4\sqrt{3} ka whakakotahi i ngā kupu rite.

$$35-6\sqrt{3}-8\left(\sqrt{3}\right)^{2}=x+\sqrt{3}y$$

Ko te pūrua o \sqrt{3} ko 3.

$$35-6\sqrt{3}-8\times 3=x+\sqrt{3}y$$

Whakareatia te -8 ki te 3, ka -24.

$$35-6\sqrt{3}-24=x+\sqrt{3}y$$

Tangohia te 24 i te 35, ka 11.

$$11-6\sqrt{3}=x+\sqrt{3}y$$

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

$$x+\sqrt{3}y=11-6\sqrt{3}$$

Tangohia te x mai i ngā taha e rua.

$$\sqrt{3}y=11-6\sqrt{3}-x$$

He hanga arowhānui tō te whārite.

$$\sqrt{3}y=-x+11-6\sqrt{3}$$

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

$$\frac{\sqrt{3}y}{\sqrt{3}}=\frac{-x+11-6\sqrt{3}}{\sqrt{3}}$$

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

$$y=\frac{-x+11-6\sqrt{3}}{\sqrt{3}}$$

Whakawehe -6\sqrt{3}-x+11 ki te \sqrt{3}.

$$y=\frac{\sqrt{3}\left(-x+11-6\sqrt{3}\right)}{3}$$

Whakaoti mō x

$$x=-\sqrt{3}y+11-6\sqrt{3}$$