Whakaoti i te frac{6+sqrt{27}}{4-sqrt{3}}

$$\frac { 6 + \sqrt { 27 } } { 4 - \sqrt { 3 } }$$

Aromātai

$$\frac{18\sqrt{3}+33}{13}\approx 4.936685734$$

Ngā Upane Otinga

Tauwehea te 27=3^{2}\times 3. Tuhia anō te pūtake rua o te hua \sqrt{3^{2}\times 3} hei hua o ngā pūtake rua \sqrt{3^{2}}\sqrt{3}. Tuhia te pūtakerua o te 3^{2}.

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

Whakangāwaritia te tauraro o \frac{6+3\sqrt{3}}{4-\sqrt{3}} mā te whakarea i te taurunga me te tauraro ki te 4+\sqrt{3}.

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

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

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

Pūrua 4. Pūrua \sqrt{3}.

$$\frac{\left(6+3\sqrt{3}\right)\left(4+\sqrt{3}\right)}{16-3}$$

Tangohia te 3 i te 16, ka 13.

$$\frac{\left(6+3\sqrt{3}\right)\left(4+\sqrt{3}\right)}{13}$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 6+3\sqrt{3} ki ia tau o 4+\sqrt{3}.

$$\frac{24+6\sqrt{3}+12\sqrt{3}+3\left(\sqrt{3}\right)^{2}}{13}$$

Pahekotia te 6\sqrt{3} me 12\sqrt{3}, ka 18\sqrt{3}.

$$\frac{24+18\sqrt{3}+3\left(\sqrt{3}\right)^{2}}{13}$$

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{24+18\sqrt{3}+3\times 3}{13}$$

Whakareatia te 3 ki te 3, ka 9.

$$\frac{24+18\sqrt{3}+9}{13}$$

Tāpirihia te 24 ki te 9, ka 33.

$$\frac{33+18\sqrt{3}}{13}$$