Réitigh frac{5-sqrt{7}}{5+sqrt{7}}+frac{5+sqrt{7}}{5-sqrt{7}}

$$\frac { 5 - \sqrt { 7 } } { 5 + \sqrt { 7 } } + \frac { 5 + \sqrt { 7 } } { 5 - \sqrt { 7 } }$$

Luacháil

$$\frac{32}{9}\approx 3.555555556$$

Céimeanna gearra réitigh

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi 5-\sqrt{7} chun ainmneoir \frac{5-\sqrt{7}}{5+\sqrt{7}} a thiontú in uimhir chóimheasta.

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

Mar shampla \left(5+\sqrt{7}\right)\left(5-\sqrt{7}\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Cearnóg 5. Cearnóg \sqrt{7}.

$$\frac{\left(5-\sqrt{7}\right)\left(5-\sqrt{7}\right)}{25-7}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Dealaigh 7 ó 25 chun 18 a fháil.

$$\frac{\left(5-\sqrt{7}\right)\left(5-\sqrt{7}\right)}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Méadaigh 5-\sqrt{7} agus 5-\sqrt{7} chun \left(5-\sqrt{7}\right)^{2} a fháil.

$$\frac{\left(5-\sqrt{7}\right)^{2}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Úsáid an teoirim dhéthéarmach \left(a-b\right)^{2}=a^{2}-2ab+b^{2} chun \left(5-\sqrt{7}\right)^{2} a leathnú.

$$\frac{25-10\sqrt{7}+\left(\sqrt{7}\right)^{2}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Is é 7 uimhir chearnach \sqrt{7}.

$$\frac{25-10\sqrt{7}+7}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Suimigh 25 agus 7 chun 32 a fháil.

$$\frac{32-10\sqrt{7}}{18}+\frac{5+\sqrt{7}}{5-\sqrt{7}}$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi 5+\sqrt{7} chun ainmneoir \frac{5+\sqrt{7}}{5-\sqrt{7}} a thiontú in uimhir chóimheasta.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{\left(5-\sqrt{7}\right)\left(5+\sqrt{7}\right)}$$

Mar shampla \left(5-\sqrt{7}\right)\left(5+\sqrt{7}\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{5^{2}-\left(\sqrt{7}\right)^{2}}$$

Cearnóg 5. Cearnóg \sqrt{7}.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{25-7}$$

Dealaigh 7 ó 25 chun 18 a fháil.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)\left(5+\sqrt{7}\right)}{18}$$

Méadaigh 5+\sqrt{7} agus 5+\sqrt{7} chun \left(5+\sqrt{7}\right)^{2} a fháil.

$$\frac{32-10\sqrt{7}}{18}+\frac{\left(5+\sqrt{7}\right)^{2}}{18}$$

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(5+\sqrt{7}\right)^{2} a leathnú.

$$\frac{32-10\sqrt{7}}{18}+\frac{25+10\sqrt{7}+\left(\sqrt{7}\right)^{2}}{18}$$

Is é 7 uimhir chearnach \sqrt{7}.

$$\frac{32-10\sqrt{7}}{18}+\frac{25+10\sqrt{7}+7}{18}$$

Suimigh 25 agus 7 chun 32 a fháil.

$$\frac{32-10\sqrt{7}}{18}+\frac{32+10\sqrt{7}}{18}$$

Tá an t-ainmneoir céanna ag \frac{32-10\sqrt{7}}{18} agus \frac{32+10\sqrt{7}}{18} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

$$\frac{32-10\sqrt{7}+32+10\sqrt{7}}{18}$$

Déan áirimh in 32-10\sqrt{7}+32+10\sqrt{7}.

$$\frac{64}{18}$$

Laghdaigh an codán \frac{64}{18} chuig na téarmaí is ísle trí 2 a bhaint agus a chealú.

$$\frac{32}{9}$$

Fachtóirigh

$$\frac{2 ^ {5}}{3 ^ {2}} = 3\frac{5}{9} = 3.5555555555555554$$