Whakaoti i te sqrt{15}div(1/sqrt{3}+1/sqrt{5})

$$\sqrt{ 15 } \div ( \frac{ 1 }{ \sqrt{ 3 } } + \frac{ 1 }{ \sqrt{ 5 } } )$$

Aromātai

$$\frac{15\left(\sqrt{5}-\sqrt{3}\right)}{2}\approx 3.780128774$$

Ngā Hipanga Rongoā Poto

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

$$\frac{\sqrt{15}}{\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{1}{\sqrt{5}}}$$

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{\sqrt{15}}{\frac{\sqrt{3}}{3}+\frac{1}{\sqrt{5}}}$$

Whakangāwaritia te tauraro o \frac{1}{\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{5}.

$$\frac{\sqrt{15}}{\frac{\sqrt{3}}{3}+\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{\sqrt{15}}{\frac{\sqrt{3}}{3}+\frac{\sqrt{5}}{5}}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 3 me 5 ko 15. Whakareatia \frac{\sqrt{3}}{3} ki te \frac{5}{5}. Whakareatia \frac{\sqrt{5}}{5} ki te \frac{3}{3}.

$$\frac{\sqrt{15}}{\frac{5\sqrt{3}}{15}+\frac{3\sqrt{5}}{15}}$$

Tā te mea he rite te tauraro o \frac{5\sqrt{3}}{15} me \frac{3\sqrt{5}}{15}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{\sqrt{15}}{\frac{5\sqrt{3}+3\sqrt{5}}{15}}$$

Whakawehe \sqrt{15} ki te \frac{5\sqrt{3}+3\sqrt{5}}{15} mā te whakarea \sqrt{15} ki te tau huripoki o \frac{5\sqrt{3}+3\sqrt{5}}{15}.

$$\frac{\sqrt{15}\times 15}{5\sqrt{3}+3\sqrt{5}}$$

Whakangāwaritia te tauraro o \frac{\sqrt{15}\times 15}{5\sqrt{3}+3\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te 5\sqrt{3}-3\sqrt{5}.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{\left(5\sqrt{3}+3\sqrt{5}\right)\left(5\sqrt{3}-3\sqrt{5}\right)}$$

Whakaarohia te \left(5\sqrt{3}+3\sqrt{5}\right)\left(5\sqrt{3}-3\sqrt{5}\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}.

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

Whakarohaina te \left(5\sqrt{3}\right)^{2}.

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

Tātaihia te 5 mā te pū o 2, kia riro ko 25.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{25\left(\sqrt{3}\right)^{2}-\left(3\sqrt{5}\right)^{2}}$$

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{25\times 3-\left(3\sqrt{5}\right)^{2}}$$

Whakareatia te 25 ki te 3, ka 75.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-\left(3\sqrt{5}\right)^{2}}$$

Whakarohaina te \left(3\sqrt{5}\right)^{2}.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-3^{2}\left(\sqrt{5}\right)^{2}}$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-9\left(\sqrt{5}\right)^{2}}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-9\times 5}$$

Whakareatia te 9 ki te 5, ka 45.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{75-45}$$

Tangohia te 45 i te 75, ka 30.

$$\frac{\sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right)}{30}$$

Whakawehea te \sqrt{15}\times 15\left(5\sqrt{3}-3\sqrt{5}\right) ki te 30, kia riro ko \sqrt{15}\times \frac{1}{2}\left(5\sqrt{3}-3\sqrt{5}\right).

$$\sqrt{15}\times \frac{1}{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \sqrt{15}\times \frac{1}{2} ki te 5\sqrt{3}-3\sqrt{5}.

$$\sqrt{15}\times \frac{1}{2}\times 5\sqrt{3}+\sqrt{15}\times \frac{1}{2}\left(-3\right)\sqrt{5}$$

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}.

$$\sqrt{3}\sqrt{5}\times \frac{1}{2}\times 5\sqrt{3}+\sqrt{15}\times \frac{1}{2}\left(-3\right)\sqrt{5}$$

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

$$3\times \frac{1}{2}\times 5\sqrt{5}+\sqrt{15}\times \frac{1}{2}\left(-3\right)\sqrt{5}$$

Whakareatia te 3 ki te \frac{1}{2}, ka \frac{3}{2}.

$$\frac{3}{2}\times 5\sqrt{5}+\sqrt{15}\times \frac{1}{2}\left(-3\right)\sqrt{5}$$

Tuhia te \frac{3}{2}\times 5 hei hautanga kotahi.

$$\frac{3\times 5}{2}\sqrt{5}+\sqrt{15}\times \frac{1}{2}\left(-3\right)\sqrt{5}$$

Whakareatia te 3 ki te 5, ka 15.

$$\frac{15}{2}\sqrt{5}+\sqrt{15}\times \frac{1}{2}\left(-3\right)\sqrt{5}$$

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

$$\frac{15}{2}\sqrt{5}+\sqrt{5}\sqrt{3}\times \frac{1}{2}\left(-3\right)\sqrt{5}$$

Whakareatia te \sqrt{5} ki te \sqrt{5}, ka 5.

$$\frac{15}{2}\sqrt{5}+5\times \frac{1}{2}\left(-3\right)\sqrt{3}$$

Whakareatia te 5 ki te \frac{1}{2}, ka \frac{5}{2}.

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

Tuhia te \frac{5}{2}\left(-3\right) hei hautanga kotahi.

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

Whakareatia te 5 ki te -3, ka -15.

$$\frac{15}{2}\sqrt{5}+\frac{-15}{2}\sqrt{3}$$

Ka taea te hautanga \frac{-15}{2} te tuhi anō ko -\frac{15}{2} mā te tango i te tohu tōraro.

$$\frac{15}{2}\sqrt{5}-\frac{15}{2}\sqrt{3}$$