Whakaoti i te (10/sqrt{5}-5/sqrt{3})(2/sqrt{3}+4/sqrt{5})

$$( \frac { 10 } { \sqrt { 5 } } - \frac { 5 } { \sqrt { 3 } } ) ( \frac { 2 } { \sqrt { 3 } } + \frac { 4 } { \sqrt { 5 } } )$$

Aromātai

$$\frac{14}{3}\approx 4.666666667$$

Ngā Hipanga Rongoā Poto

Whakangāwaritia te tauraro o \frac{10}{\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{5}.

$$\left(\frac{10\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{5}{\sqrt{3}}\right)\left(\frac{2}{\sqrt{3}}+\frac{4}{\sqrt{5}}\right)$$

Ko te pūrua o \sqrt{5} ko 5.

$$\left(\frac{10\sqrt{5}}{5}-\frac{5}{\sqrt{3}}\right)\left(\frac{2}{\sqrt{3}}+\frac{4}{\sqrt{5}}\right)$$

Whakawehea te 10\sqrt{5} ki te 5, kia riro ko 2\sqrt{5}.

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

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

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

Ko te pūrua o \sqrt{3} ko 3.

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

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 2\sqrt{5} ki te \frac{3}{3}.

$$\left(\frac{3\times 2\sqrt{5}}{3}-\frac{5\sqrt{3}}{3}\right)\left(\frac{2}{\sqrt{3}}+\frac{4}{\sqrt{5}}\right)$$

Tā te mea he rite te tauraro o \frac{3\times 2\sqrt{5}}{3} me \frac{5\sqrt{3}}{3}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{3\times 2\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2}{\sqrt{3}}+\frac{4}{\sqrt{5}}\right)$$

Mahia ngā whakarea i roto o 3\times 2\sqrt{5}-5\sqrt{3}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2}{\sqrt{3}}+\frac{4}{\sqrt{5}}\right)$$

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

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{4}{\sqrt{5}}\right)$$

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{3}+\frac{4}{\sqrt{5}}\right)$$

Whakangāwaritia te tauraro o \frac{4}{\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{5}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{3}+\frac{4\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\right)$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{2\sqrt{3}}{3}+\frac{4\sqrt{5}}{5}\right)$$

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{2\sqrt{3}}{3} ki te \frac{5}{5}. Whakareatia \frac{4\sqrt{5}}{5} ki te \frac{3}{3}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\left(\frac{5\times 2\sqrt{3}}{15}+\frac{3\times 4\sqrt{5}}{15}\right)$$

Tā te mea he rite te tauraro o \frac{5\times 2\sqrt{3}}{15} me \frac{3\times 4\sqrt{5}}{15}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\times \frac{5\times 2\sqrt{3}+3\times 4\sqrt{5}}{15}$$

Mahia ngā whakarea i roto o 5\times 2\sqrt{3}+3\times 4\sqrt{5}.

$$\frac{6\sqrt{5}-5\sqrt{3}}{3}\times \frac{10\sqrt{3}+12\sqrt{5}}{15}$$

Me whakarea te \frac{6\sqrt{5}-5\sqrt{3}}{3} ki te \frac{10\sqrt{3}+12\sqrt{5}}{15} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\left(6\sqrt{5}-5\sqrt{3}\right)\left(10\sqrt{3}+12\sqrt{5}\right)}{3\times 15}$$

Whakareatia te 3 ki te 15, ka 45.

$$\frac{\left(6\sqrt{5}-5\sqrt{3}\right)\left(10\sqrt{3}+12\sqrt{5}\right)}{45}$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 6\sqrt{5}-5\sqrt{3} ki ia tau o 10\sqrt{3}+12\sqrt{5}.

$$\frac{60\sqrt{3}\sqrt{5}+72\left(\sqrt{5}\right)^{2}-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

Hei whakarea \sqrt{3} me \sqrt{5}, whakareatia ngā tau i raro i te pūtake rua.

$$\frac{60\sqrt{15}+72\left(\sqrt{5}\right)^{2}-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{60\sqrt{15}+72\times 5-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

Whakareatia te 72 ki te 5, ka 360.

$$\frac{60\sqrt{15}+360-50\left(\sqrt{3}\right)^{2}-60\sqrt{3}\sqrt{5}}{45}$$

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{60\sqrt{15}+360-50\times 3-60\sqrt{3}\sqrt{5}}{45}$$

Whakareatia te -50 ki te 3, ka -150.

$$\frac{60\sqrt{15}[phone]\sqrt{3}\sqrt{5}}{45}$$

Tangohia te 150 i te 360, ka 210.

$$\frac{60\sqrt{15}+210-60\sqrt{3}\sqrt{5}}{45}$$

Hei whakarea \sqrt{3} me \sqrt{5}, whakareatia ngā tau i raro i te pūtake rua.

$$\frac{60\sqrt{15}+210-60\sqrt{15}}{45}$$

Pahekotia te 60\sqrt{15} me -60\sqrt{15}, ka 0.

$$\frac{210}{45}$$

Whakahekea te hautanga \frac{210}{45} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 15.

$$\frac{14}{3}$$

Tauwehe

$$\frac{2 \cdot 7}{3} = 4\frac{2}{3} = 4.666666666666667$$