Réitigh frac{31sqrt{2}+31sqrt{5}}{2sqrt{10}-3}-frac{62sqrt{2}}{3-2sqrt{10}}

$$\frac { 31 \sqrt { 2 } + 31 \sqrt { 5 } } { 2 \sqrt { 10 } - 3 } - \frac { 62 \sqrt { 2 } } { 3 - 2 \sqrt { 10 } }$$

Luacháil

$$15\sqrt{5}+19\sqrt{2}\approx 60.411077348$$

Céimeanna Réitigh

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi 2\sqrt{10}+3 chun ainmneoir \frac{31\sqrt{2}+31\sqrt{5}}{2\sqrt{10}-3} a thiontú in uimhir chóimheasta.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{\left(2\sqrt{10}-3\right)\left(2\sqrt{10}+3\right)}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Mar shampla \left(2\sqrt{10}-3\right)\left(2\sqrt{10}+3\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}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{\left(2\sqrt{10}\right)^{2}-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Fairsingigh \left(2\sqrt{10}\right)^{2}

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{2^{2}\left(\sqrt{10}\right)^{2}-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Ríomh cumhacht 2 de 2 agus faigh 4.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{4\left(\sqrt{10}\right)^{2}-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Is é 10 uimhir chearnach \sqrt{10}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{4\times 10-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Méadaigh 4 agus 10 chun 40 a fháil.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{40-3^{2}}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Ríomh cumhacht 3 de 2 agus faigh 9.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{40-9}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Dealaigh 9 ó 40 chun 31 a fháil.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}}{3-2\sqrt{10}}$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi 3+2\sqrt{10} chun ainmneoir \frac{62\sqrt{2}}{3-2\sqrt{10}} a thiontú in uimhir chóimheasta.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{\left(3-2\sqrt{10}\right)\left(3+2\sqrt{10}\right)}$$

Mar shampla \left(3-2\sqrt{10}\right)\left(3+2\sqrt{10}\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}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{3^{2}-\left(-2\sqrt{10}\right)^{2}}$$

Ríomh cumhacht 3 de 2 agus faigh 9.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-\left(-2\sqrt{10}\right)^{2}}$$

Fairsingigh \left(-2\sqrt{10}\right)^{2}

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-\left(-2\right)^{2}\left(\sqrt{10}\right)^{2}}$$

Ríomh cumhacht -2 de 2 agus faigh 4.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-4\left(\sqrt{10}\right)^{2}}$$

Is é 10 uimhir chearnach \sqrt{10}.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-4\times 10}$$

Méadaigh 4 agus 10 chun 40 a fháil.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{9-40}$$

Dealaigh 40 ó 9 chun -31 a fháil.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\frac{62\sqrt{2}\left(3+2\sqrt{10}\right)}{-31}$$

Roinn 62\sqrt{2}\left(3+2\sqrt{10}\right) faoi -31 chun -2\sqrt{2}\left(3+2\sqrt{10}\right) a fháil.

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}-\left(-2\sqrt{2}\left(3+2\sqrt{10}\right)\right)$$

Tá 2\sqrt{2}\left(3+2\sqrt{10}\right) urchomhairleach le -2\sqrt{2}\left(3+2\sqrt{10}\right).

$$\frac{\left(31\sqrt{2}+31\sqrt{5}\right)\left(2\sqrt{10}+3\right)}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Cuir an t-airí dáileacháin i bhfeidhm trí gach téarma de 31\sqrt{2}+31\sqrt{5} a iolrú faoi gach téarma de 2\sqrt{10}+3.

$$\frac{62\sqrt{10}\sqrt{2}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Fachtóirigh 10=2\times 5. Athscríobh fréamh cearnach an toraidh \sqrt{2\times 5} mar thoradh na bhfréamhacha cearnacha \sqrt{2}\sqrt{5}.

$$\frac{62\sqrt{2}\sqrt{5}\sqrt{2}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Méadaigh \sqrt{2} agus \sqrt{2} chun 2 a fháil.

$$\frac{62\times 2\sqrt{5}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Méadaigh 62 agus 2 chun 124 a fháil.

$$\frac{124\sqrt{5}+93\sqrt{2}+62\sqrt{5}\sqrt{10}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Fachtóirigh 10=5\times 2. Athscríobh fréamh cearnach an toraidh \sqrt{5\times 2} mar thoradh na bhfréamhacha cearnacha \sqrt{5}\sqrt{2}.

$$\frac{124\sqrt{5}+93\sqrt{2}+62\sqrt{5}\sqrt{5}\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Méadaigh \sqrt{5} agus \sqrt{5} chun 5 a fháil.

$$\frac{124\sqrt{5}+93\sqrt{2}+62\times 5\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Méadaigh 62 agus 5 chun 310 a fháil.

$$\frac{124\sqrt{5}+93\sqrt{2}+310\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Comhcheangail 93\sqrt{2} agus 310\sqrt{2} chun 403\sqrt{2} a fháil.

$$\frac{124\sqrt{5}+403\sqrt{2}+93\sqrt{5}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Comhcheangail 124\sqrt{5} agus 93\sqrt{5} chun 217\sqrt{5} a fháil.

$$\frac{217\sqrt{5}+403\sqrt{2}}{31}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Roinn 217\sqrt{5}+403\sqrt{2} faoi 31 chun 7\sqrt{5}+13\sqrt{2} a fháil.

$$7\sqrt{5}+13\sqrt{2}+2\sqrt{2}\left(3+2\sqrt{10}\right)$$

Úsáid an t-airí dáileach chun 2\sqrt{2} a mhéadú faoi 3+2\sqrt{10}.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+4\sqrt{10}\sqrt{2}$$

Fachtóirigh 10=2\times 5. Athscríobh fréamh cearnach an toraidh \sqrt{2\times 5} mar thoradh na bhfréamhacha cearnacha \sqrt{2}\sqrt{5}.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+4\sqrt{2}\sqrt{5}\sqrt{2}$$

Méadaigh \sqrt{2} agus \sqrt{2} chun 2 a fháil.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+4\times 2\sqrt{5}$$

Méadaigh 4 agus 2 chun 8 a fháil.

$$7\sqrt{5}+13\sqrt{2}+6\sqrt{2}+8\sqrt{5}$$

Comhcheangail 13\sqrt{2} agus 6\sqrt{2} chun 19\sqrt{2} a fháil.

$$7\sqrt{5}+19\sqrt{2}+8\sqrt{5}$$

Comhcheangail 7\sqrt{5} agus 8\sqrt{5} chun 15\sqrt{5} a fháil.

$$15\sqrt{5}+19\sqrt{2}$$

Fachtóirigh

$$15 \sqrt{5} + 19 \sqrt{2} = 60.411077348$$