Whakaoti i te frac{(3+sqrt{5})^2-(2-sqrt{5})(2+sqrt{5})}{(sqrt{7}-2)(sqrt{7}+2)}

$$\frac { ( 3 + \sqrt { 5 } ) ^ { 2 } - ( 2 - \sqrt { 5 } ) ( 2 + \sqrt { 5 } ) } { ( \sqrt { 7 } - 2 ) ( \sqrt { 7 } + 2 ) }$$

Aromātai

$$2\sqrt{5}+5\approx 9.472135955$$

Ngā Upane Otinga

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

$$\frac{9+6\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{9+6\sqrt{5}+5-\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Tāpirihia te 9 ki te 5, ka 14.

$$\frac{14+6\sqrt{5}-\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Whakaarohia te \left(2-\sqrt{5}\right)\left(2+\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}. Pūrua 2.

$$\frac{14+6\sqrt{5}-\left(4-\left(\sqrt{5}\right)^{2}\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{14+6\sqrt{5}-\left(4-5\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Tangohia te 5 i te 4, ka -1.

$$\frac{14+6\sqrt{5}-\left(-1\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Ko te tauaro o -1 ko 1.

$$\frac{14+6\sqrt{5}+1}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}$$

Whakaarohia te \left(\sqrt{7}-2\right)\left(\sqrt{7}+2\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}. Pūrua 2.

$$\frac{14+6\sqrt{5}+1}{\left(\sqrt{7}\right)^{2}-4}$$

Ko te pūrua o \sqrt{7} ko 7.

$$\frac{14+6\sqrt{5}+1}{7-4}$$

Tangohia te 4 i te 7, ka 3.

$$\frac{14+6\sqrt{5}+1}{3}$$

Tāpirihia te 14 ki te 1, ka 15.

$$\frac{15+6\sqrt{5}}{3}$$

Whakawehea ia wā o 15+6\sqrt{5} ki te 3, kia riro ko 5+2\sqrt{5}.

$$5+2\sqrt{5}$$