Atrisiniet frac{(sqrt{5}-sqrt{2})(3sqrt{5}+sqrt{2})}{3sqrt{5}+2sqrt{2}}

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

Izrēķināt

$$\frac{47\sqrt{5}-56\sqrt{2}}{37}\approx 0,699979336$$

Atrisinājuma darbības

Atbrīvojieties no iracionalitātes saucēju ar \frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)}{3\sqrt{5}+2\sqrt{2}}, reizinot skaitītāju un saucēju ar 3\sqrt{5}-2\sqrt{2}.

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

Apsveriet \left(3\sqrt{5}+2\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right). Reizināšanu var pārvērst par kvadrātu starpību, izmantojot šo kārtulu: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Paplašiniet \left(3\sqrt{5}\right)^{2}.

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

Aprēķiniet 3 pakāpē 2 un iegūstiet 9.

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

Skaitļa \sqrt{5} kvadrāts ir 5.

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

Reiziniet 9 un 5, lai iegūtu 45.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-\left(2\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-2^{2}\left(\sqrt{2}\right)^{2}}$$

Aprēķiniet 2 pakāpē 2 un iegūstiet 4.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-4\left(\sqrt{2}\right)^{2}}$$

Skaitļa \sqrt{2} kvadrāts ir 2.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-4\times 2}$$

Reiziniet 4 un 2, lai iegūtu 8.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{45-8}$$

Atņemiet 8 no 45, lai iegūtu 37.

$$\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

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

$$\frac{\left(3\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Skaitļa \sqrt{5} kvadrāts ir 5.

$$\frac{\left(3\times 5+\sqrt{5}\sqrt{2}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Reiziniet 3 un 5, lai iegūtu 15.

$$\frac{\left(15+\sqrt{5}\sqrt{2}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Lai reiziniet \sqrt{5} un \sqrt{2}, reiziniet numurus zem kvadrātveida saknes.

$$\frac{\left(15+\sqrt{10}-3\sqrt{2}\sqrt{5}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Lai reiziniet \sqrt{2} un \sqrt{5}, reiziniet numurus zem kvadrātveida saknes.

$$\frac{\left(15+\sqrt{10}-3\sqrt{10}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Savelciet \sqrt{10} un -3\sqrt{10}, lai iegūtu -2\sqrt{10}.

$$\frac{\left(15-2\sqrt{10}-\left(\sqrt{2}\right)^{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Skaitļa \sqrt{2} kvadrāts ir 2.

$$\frac{\left(15-2\sqrt{10}-2\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Atņemiet 2 no 15, lai iegūtu 13.

$$\frac{\left(13-2\sqrt{10}\right)\left(3\sqrt{5}-2\sqrt{2}\right)}{37}$$

Izmantojiet distributīvo īpašību, katru 13-2\sqrt{10} locekli reizinot ar katru 3\sqrt{5}-2\sqrt{2} locekli.

$$\frac{39\sqrt{5}-26\sqrt{2}-6\sqrt{10}\sqrt{5}+4\sqrt{2}\sqrt{10}}{37}$$

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

$$\frac{39\sqrt{5}-26\sqrt{2}-6\sqrt{5}\sqrt{2}\sqrt{5}+4\sqrt{2}\sqrt{10}}{37}$$

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

$$\frac{39\sqrt{5}-26\sqrt{2}-6\times 5\sqrt{2}+4\sqrt{2}\sqrt{10}}{37}$$

Reiziniet -6 un 5, lai iegūtu -30.

$$\frac{39\sqrt{5}-26\sqrt{2}-30\sqrt{2}+4\sqrt{2}\sqrt{10}}{37}$$

Savelciet -26\sqrt{2} un -30\sqrt{2}, lai iegūtu -56\sqrt{2}.

$$\frac{39\sqrt{5}-56\sqrt{2}+4\sqrt{2}\sqrt{10}}{37}$$

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

$$\frac{39\sqrt{5}-56\sqrt{2}+4\sqrt{2}\sqrt{2}\sqrt{5}}{37}$$

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

$$\frac{39\sqrt{5}-56\sqrt{2}+4\times 2\sqrt{5}}{37}$$

Reiziniet 4 un 2, lai iegūtu 8.

$$\frac{39\sqrt{5}-56\sqrt{2}+8\sqrt{5}}{37}$$

Savelciet 39\sqrt{5} un 8\sqrt{5}, lai iegūtu 47\sqrt{5}.

$$\frac{47\sqrt{5}-56\sqrt{2}}{37}$$