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

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

Ovrednoti

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

Koraki rešitve

Racionalizirajte imenovalec \frac{\left(\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)}{3\sqrt{5}+2\sqrt{2}} tako, da pomnožite števec in imenovalec s 3\sqrt{5}-2\sqrt{2}.

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

Razmislite o \left(3\sqrt{5}+2\sqrt{2}\right)\left(3\sqrt{5}-2\sqrt{2}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Izračunajte potenco 3 števila 2, da dobite 9.

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

Kvadrat vrednosti \sqrt{5} je 5.

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

Pomnožite 9 in 5, da dobite 45.

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

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

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

Izračunajte potenco 2 števila 2, da dobite 4.

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

Kvadrat vrednosti \sqrt{2} je 2.

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

Pomnožite 4 in 2, da dobite 8.

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

Odštejte 8 od 45, da dobite 37.

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

Uporabite distributivnost tako, da pomnožite vsako vrednost \sqrt{5}-\sqrt{2} z vsako vrednostjo 3\sqrt{5}+\sqrt{2}.

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

Kvadrat vrednosti \sqrt{5} je 5.

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

Pomnožite 3 in 5, da dobite 15.

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

Če želite \sqrt{5} pomnožite in \sqrt{2}, pomnožite številke v kvadratni korenu.

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

Če želite \sqrt{2} pomnožite in \sqrt{5}, pomnožite številke v kvadratni korenu.

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

Združite \sqrt{10} in -3\sqrt{10}, da dobite -2\sqrt{10}.

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

Kvadrat vrednosti \sqrt{2} je 2.

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

Odštejte 2 od 15, da dobite 13.

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

Uporabite distributivnost tako, da pomnožite vsako vrednost 13-2\sqrt{10} z vsako vrednostjo 3\sqrt{5}-2\sqrt{2}.

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

Faktorizirajte 10=5\times 2. Znova napišite kvadratni koren izdelka \sqrt{5\times 2} kot produkt kvadratnih korenov \sqrt{5}\sqrt{2}.

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

Pomnožite \sqrt{5} in \sqrt{5}, da dobite 5.

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

Pomnožite -6 in 5, da dobite -30.

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

Združite -26\sqrt{2} in -30\sqrt{2}, da dobite -56\sqrt{2}.

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

Faktorizirajte 10=2\times 5. Znova napišite kvadratni koren izdelka \sqrt{2\times 5} kot produkt kvadratnih korenov \sqrt{2}\sqrt{5}.

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

Pomnožite \sqrt{2} in \sqrt{2}, da dobite 2.

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

Pomnožite 4 in 2, da dobite 8.

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

Združite 39\sqrt{5} in 8\sqrt{5}, da dobite 47\sqrt{5}.

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