Whakaoti i te frac{sqrt{5}+sqrt{3}}{sqrt{5}-sqrt{3}}-frac{sqrt{5}-sqrt{3}}{sqrt{5}+sqrt{3}}=2sqrt{15}

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

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

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

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

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

Pūrua \sqrt{5}. Pūrua \sqrt{3}.

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

Tangohia te 3 i te 5, ka 2.

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

Whakareatia te \sqrt{5}+\sqrt{3} ki te \sqrt{5}+\sqrt{3}, ka \left(\sqrt{5}+\sqrt{3}\right)^{2}.

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(\sqrt{5}+\sqrt{3}\right)^{2}.

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

Ko te pūrua o \sqrt{5} ko 5.

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

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

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

Ko te pūrua o \sqrt{3} ko 3.

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

Tāpirihia te 5 ki te 3, ka 8.

$$\frac{8+2\sqrt{15}}{2}-\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}=2\sqrt{15}$$

Whakawehea ia wā o 8+2\sqrt{15} ki te 2, kia riro ko 4+\sqrt{15}.

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

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

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

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

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

Pūrua \sqrt{5}. Pūrua \sqrt{3}.

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

Tangohia te 3 i te 5, ka 2.

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

Whakareatia te \sqrt{5}-\sqrt{3} ki te \sqrt{5}-\sqrt{3}, ka \left(\sqrt{5}-\sqrt{3}\right)^{2}.

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

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{5}-\sqrt{3}\right)^{2}.

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

Ko te pūrua o \sqrt{5} ko 5.

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

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

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

Ko te pūrua o \sqrt{3} ko 3.

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

Tāpirihia te 5 ki te 3, ka 8.

$$4+\sqrt{15}-\frac{8-2\sqrt{15}}{2}=2\sqrt{15}$$

Whakawehea ia wā o 8-2\sqrt{15} ki te 2, kia riro ko 4-\sqrt{15}.

$$4+\sqrt{15}-\left(4-\sqrt{15}\right)=2\sqrt{15}$$

Hei kimi i te tauaro o 4-\sqrt{15}, kimihia te tauaro o ia taurangi.

$$4+\sqrt{15}-4-\left(-\sqrt{15}\right)=2\sqrt{15}$$

Ko te tauaro o -\sqrt{15} ko \sqrt{15}.

$$4+\sqrt{15}-4+\sqrt{15}=2\sqrt{15}$$

Tangohia te 4 i te 4, ka 0.

$$\sqrt{15}+\sqrt{15}=2\sqrt{15}$$

Pahekotia te \sqrt{15} me \sqrt{15}, ka 2\sqrt{15}.

$$2\sqrt{15}=2\sqrt{15}$$

Tangohia te 2\sqrt{15} mai i ngā taha e rua.

$$2\sqrt{15}-2\sqrt{15}=0$$

Pahekotia te 2\sqrt{15} me -2\sqrt{15}, ka 0.

$$0=0$$

Whakatauritea te 0 me te 0.

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