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

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

Arvuta

$$15\sqrt{5}+19\sqrt{2}\approx 60,411077348$$

Lahenduse etapid

Ratsionaliseerige korrutades lugeja ja 2\sqrt{10}+3 nimetaja \frac{31\sqrt{2}+31\sqrt{5}}{2\sqrt{10}-3} nimetaja.

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

Mõelge valemile \left(2\sqrt{10}-3\right)\left(2\sqrt{10}+3\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Laiendage \left(2\sqrt{10}\right)^{2}.

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

Arvutage 2 aste 2 ja leidke 4.

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

\sqrt{10} ruut on 10.

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

Korrutage 4 ja 10, et leida 40.

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

Arvutage 2 aste 3 ja leidke 9.

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

Lahutage 9 väärtusest 40, et leida 31.

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

Ratsionaliseerige korrutades lugeja ja 3+2\sqrt{10} nimetaja \frac{62\sqrt{2}}{3-2\sqrt{10}} nimetaja.

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

Mõelge valemile \left(3-2\sqrt{10}\right)\left(3+2\sqrt{10}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Arvutage 2 aste 3 ja leidke 9.

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

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

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

Arvutage 2 aste -2 ja leidke 4.

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

\sqrt{10} ruut on 10.

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

Korrutage 4 ja 10, et leida 40.

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

Lahutage 40 väärtusest 9, et leida -31.

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

Jagage 62\sqrt{2}\left(3+2\sqrt{10}\right) väärtusega -31, et leida -2\sqrt{2}\left(3+2\sqrt{10}\right).

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

Arvu -2\sqrt{2}\left(3+2\sqrt{10}\right) vastand on 2\sqrt{2}\left(3+2\sqrt{10}\right).

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

Rakendage distributiivsusomadus, korrutades avaldise 31\sqrt{2}+31\sqrt{5} iga liikme avaldise 2\sqrt{10}+3 iga liikmega.

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

Tegurda 10=2\times 5. Kirjutage \sqrt{2\times 5} toote juured, kui see ruut \sqrt{2}\sqrt{5}.

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

Korrutage \sqrt{2} ja \sqrt{2}, et leida 2.

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

Korrutage 62 ja 2, et leida 124.

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

Tegurda 10=5\times 2. Kirjutage \sqrt{5\times 2} toote juured, kui see ruut \sqrt{5}\sqrt{2}.

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

Korrutage \sqrt{5} ja \sqrt{5}, et leida 5.

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

Korrutage 62 ja 5, et leida 310.

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

Kombineerige 93\sqrt{2} ja 310\sqrt{2}, et leida 403\sqrt{2}.

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

Kombineerige 124\sqrt{5} ja 93\sqrt{5}, et leida 217\sqrt{5}.

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

Jagage 217\sqrt{5}+403\sqrt{2} iga liige 31-ga, et saada 7\sqrt{5}+13\sqrt{2}.

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

Kasutage distributiivsusomadust, et korrutada 2\sqrt{2} ja 3+2\sqrt{10}.

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

Tegurda 10=2\times 5. Kirjutage \sqrt{2\times 5} toote juured, kui see ruut \sqrt{2}\sqrt{5}.

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

Korrutage \sqrt{2} ja \sqrt{2}, et leida 2.

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

Korrutage 4 ja 2, et leida 8.

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

Kombineerige 13\sqrt{2} ja 6\sqrt{2}, et leida 19\sqrt{2}.

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

Kombineerige 7\sqrt{5} ja 8\sqrt{5}, et leida 15\sqrt{5}.

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

Lahuta teguriteks

$$15 \sqrt{5} + 19 \sqrt{2} = 60,411077348$$