Whakaoti i te frac{sqrt{10}+sqrt{15}}{sqrt{2}+sqrt{3}}

$$\frac { \sqrt { 10 } + \sqrt { 15 } } { \sqrt { 2 } + \sqrt { 3 } }$$

Aromātai

$$\sqrt{5}\approx 2.236067977$$

Ngā Upane Otinga

Whakangāwaritia te tauraro o \frac{\sqrt{10}+\sqrt{15}}{\sqrt{2}+\sqrt{3}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{2}-\sqrt{3}.

$$\frac{\left(\sqrt{10}+\sqrt{15}\right)\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}$$

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

$$\frac{\left(\sqrt{10}+\sqrt{15}\right)\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Pūrua \sqrt{2}. Pūrua \sqrt{3}.

$$\frac{\left(\sqrt{10}+\sqrt{15}\right)\left(\sqrt{2}-\sqrt{3}\right)}{2-3}$$

Tangohia te 3 i te 2, ka -1.

$$\frac{\left(\sqrt{10}+\sqrt{15}\right)\left(\sqrt{2}-\sqrt{3}\right)}{-1}$$

Ko te mea whakawehea ki te -1 ka hōmai i tōna kōaro.

$$-\left(\sqrt{10}+\sqrt{15}\right)\left(\sqrt{2}-\sqrt{3}\right)$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o \sqrt{10}+\sqrt{15} ki ia tau o \sqrt{2}-\sqrt{3}.

$$-\left(\sqrt{10}\sqrt{2}-\sqrt{10}\sqrt{3}+\sqrt{15}\sqrt{2}-\sqrt{15}\sqrt{3}\right)$$

Tauwehea te 10=2\times 5. Tuhia anō te pūtake rua o te hua \sqrt{2\times 5} hei hua o ngā pūtake rua \sqrt{2}\sqrt{5}.

$$-\left(\sqrt{2}\sqrt{5}\sqrt{2}-\sqrt{10}\sqrt{3}+\sqrt{15}\sqrt{2}-\sqrt{15}\sqrt{3}\right)$$

Whakareatia te \sqrt{2} ki te \sqrt{2}, ka 2.

$$-\left(2\sqrt{5}-\sqrt{10}\sqrt{3}+\sqrt{15}\sqrt{2}-\sqrt{15}\sqrt{3}\right)$$

Hei whakarea \sqrt{10} me \sqrt{3}, whakareatia ngā tau i raro i te pūtake rua.

$$-\left(2\sqrt{5}-\sqrt{30}+\sqrt{15}\sqrt{2}-\sqrt{15}\sqrt{3}\right)$$

Hei whakarea \sqrt{15} me \sqrt{2}, whakareatia ngā tau i raro i te pūtake rua.

$$-\left(2\sqrt{5}-\sqrt{30}+\sqrt{30}-\sqrt{15}\sqrt{3}\right)$$

Pahekotia te -\sqrt{30} me \sqrt{30}, ka 0.

$$-\left(2\sqrt{5}-\sqrt{15}\sqrt{3}\right)$$

Tauwehea te 15=3\times 5. Tuhia anō te pūtake rua o te hua \sqrt{3\times 5} hei hua o ngā pūtake rua \sqrt{3}\sqrt{5}.

$$-\left(2\sqrt{5}-\sqrt{3}\sqrt{5}\sqrt{3}\right)$$

Whakareatia te \sqrt{3} ki te \sqrt{3}, ka 3.

$$-\left(2\sqrt{5}-3\sqrt{5}\right)$$

Pahekotia te 2\sqrt{5} me -3\sqrt{5}, ka -\sqrt{5}.

$$-\left(-\sqrt{5}\right)$$

Ko te tauaro o -\sqrt{5} ko \sqrt{5}.

$$\sqrt{5}$$