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

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

Izrēķināt

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

Atrisinājuma darbības

Atbrīvojieties no iracionalitātes saucēju ar \frac{31\sqrt{2}+31\sqrt{5}}{2\sqrt{10}-3}, reizinot skaitītāju un saucēju ar 2\sqrt{10}+3.

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

Apsveriet \left(2\sqrt{10}-3\right)\left(2\sqrt{10}+3\right). Reizināšanu var pārvērst par kvadrātu starpību, izmantojot šo kārtulu: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Aprēķiniet 2 pakāpē 2 un iegūstiet 4.

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

Skaitļa \sqrt{10} kvadrāts ir 10.

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

Reiziniet 4 un 10, lai iegūtu 40.

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

Aprēķiniet 3 pakāpē 2 un iegūstiet 9.

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

Atņemiet 9 no 40, lai iegūtu 31.

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

Atbrīvojieties no iracionalitātes saucēju ar \frac{62\sqrt{2}}{3-2\sqrt{10}}, reizinot skaitītāju un saucēju ar 3+2\sqrt{10}.

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

Apsveriet \left(3-2\sqrt{10}\right)\left(3+2\sqrt{10}\right). Reizināšanu var pārvērst par kvadrātu starpību, izmantojot šo kārtulu: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Aprēķiniet 3 pakāpē 2 un iegūstiet 9.

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

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

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

Aprēķiniet -2 pakāpē 2 un iegūstiet 4.

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

Skaitļa \sqrt{10} kvadrāts ir 10.

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

Reiziniet 4 un 10, lai iegūtu 40.

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

Atņemiet 40 no 9, lai iegūtu -31.

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

Daliet 62\sqrt{2}\left(3+2\sqrt{10}\right) ar -31, lai iegūtu -2\sqrt{2}\left(3+2\sqrt{10}\right).

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

Skaitļa -2\sqrt{2}\left(3+2\sqrt{10}\right) pretstats ir 2\sqrt{2}\left(3+2\sqrt{10}\right).

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

Izmantojiet distributīvo īpašību, katru 31\sqrt{2}+31\sqrt{5} locekli reizinot ar katru 2\sqrt{10}+3 locekli.

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

Sadaliet reizinātājos 10=2\times 5. Pārrakstiet reizinājuma kvadrātsakni \sqrt{2\times 5} kā kvadrātveida saknes \sqrt{2}\sqrt{5}.

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

Reiziniet \sqrt{2} un \sqrt{2}, lai iegūtu 2.

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

Reiziniet 62 un 2, lai iegūtu 124.

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

Sadaliet reizinātājos 10=5\times 2. Pārrakstiet reizinājuma kvadrātsakni \sqrt{5\times 2} kā kvadrātveida saknes \sqrt{5}\sqrt{2}.

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

Reiziniet \sqrt{5} un \sqrt{5}, lai iegūtu 5.

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

Reiziniet 62 un 5, lai iegūtu 310.

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

Savelciet 93\sqrt{2} un 310\sqrt{2}, lai iegūtu 403\sqrt{2}.

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

Savelciet 124\sqrt{5} un 93\sqrt{5}, lai iegūtu 217\sqrt{5}.

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

Daliet katru 217\sqrt{5}+403\sqrt{2} locekli ar 31, lai iegūtu 7\sqrt{5}+13\sqrt{2}.

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

Izmantojiet distributīvo īpašību, lai reizinātu 2\sqrt{2} ar 3+2\sqrt{10}.

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

Sadaliet reizinātājos 10=2\times 5. Pārrakstiet reizinājuma kvadrātsakni \sqrt{2\times 5} kā kvadrātveida saknes \sqrt{2}\sqrt{5}.

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

Reiziniet \sqrt{2} un \sqrt{2}, lai iegūtu 2.

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

Reiziniet 4 un 2, lai iegūtu 8.

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

Savelciet 13\sqrt{2} un 6\sqrt{2}, lai iegūtu 19\sqrt{2}.

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

Savelciet 7\sqrt{5} un 8\sqrt{5}, lai iegūtu 15\sqrt{5}.

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

Sadalīt reizinātājos

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