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

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

Procijeni

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

Koraci za rješenje

Racionalizirajte imenilac broja \frac{31\sqrt{2}+31\sqrt{5}}{2\sqrt{10}-3} tako što ćete pomnožiti brojilac i imenilac sa 2\sqrt{10}+3.

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

Razmotrite \left(2\sqrt{10}-3\right)\left(2\sqrt{10}+3\right). Množenje se može transformirati u razliku kvadrata pomoću pravila: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Izračunajte 2 stepen od 2 i dobijte 4.

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

Kvadrat broja \sqrt{10} je 10.

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

Pomnožite 4 i 10 da biste dobili 40.

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

Izračunajte 3 stepen od 2 i dobijte 9.

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

Oduzmite 9 od 40 da biste dobili 31.

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

Racionalizirajte imenilac broja \frac{62\sqrt{2}}{3-2\sqrt{10}} tako što ćete pomnožiti brojilac i imenilac sa 3+2\sqrt{10}.

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

Razmotrite \left(3-2\sqrt{10}\right)\left(3+2\sqrt{10}\right). Množenje se može transformirati u razliku kvadrata pomoću pravila: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Izračunajte 3 stepen od 2 i dobijte 9.

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

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

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

Izračunajte -2 stepen od 2 i dobijte 4.

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

Kvadrat broja \sqrt{10} je 10.

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

Pomnožite 4 i 10 da biste dobili 40.

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

Oduzmite 40 od 9 da biste dobili -31.

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

Podijelite 62\sqrt{2}\left(3+2\sqrt{10}\right) sa -31 da biste dobili -2\sqrt{2}\left(3+2\sqrt{10}\right).

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

Opozit broja -2\sqrt{2}\left(3+2\sqrt{10}\right) je 2\sqrt{2}\left(3+2\sqrt{10}\right).

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

Primijenite distributivno svojstvo tako što ćete pomnožiti svaki izraz od 31\sqrt{2}+31\sqrt{5} svakim izrazom od 2\sqrt{10}+3.

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

Faktorirajte 10=2\times 5. Ponovo napišite kvadratni korijen proizvoda \sqrt{2\times 5} kao proizvod kvadratnih korijena \sqrt{2}\sqrt{5}.

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

Pomnožite \sqrt{2} i \sqrt{2} da biste dobili 2.

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

Pomnožite 62 i 2 da biste dobili 124.

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

Faktorirajte 10=5\times 2. Ponovo napišite kvadratni korijen proizvoda \sqrt{5\times 2} kao proizvod kvadratnih korijena \sqrt{5}\sqrt{2}.

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

Pomnožite \sqrt{5} i \sqrt{5} da biste dobili 5.

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

Pomnožite 62 i 5 da biste dobili 310.

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

Kombinirajte 93\sqrt{2} i 310\sqrt{2} da biste dobili 403\sqrt{2}.

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

Kombinirajte 124\sqrt{5} i 93\sqrt{5} da biste dobili 217\sqrt{5}.

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

Podijelite svaki element izraza 217\sqrt{5}+403\sqrt{2} s 31 da biste dobili 7\sqrt{5}+13\sqrt{2}.

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

Koristite distributivno svojstvo da biste pomnožili 2\sqrt{2} sa 3+2\sqrt{10}.

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

Faktorirajte 10=2\times 5. Ponovo napišite kvadratni korijen proizvoda \sqrt{2\times 5} kao proizvod kvadratnih korijena \sqrt{2}\sqrt{5}.

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

Pomnožite \sqrt{2} i \sqrt{2} da biste dobili 2.

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

Pomnožite 4 i 2 da biste dobili 8.

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

Kombinirajte 13\sqrt{2} i 6\sqrt{2} da biste dobili 19\sqrt{2}.

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

Kombinirajte 7\sqrt{5} i 8\sqrt{5} da biste dobili 15\sqrt{5}.

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

Faktor

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