Lahenda frac{sqrt{18}-sqrt{12}}{sqrt{50}-sqrt{48}}

$$\frac{ \sqrt{ 18 } - \sqrt{ 12 } }{ \sqrt{ 50 } - \sqrt{ 48 } }$$

Arvuta

$$\sqrt{6}+3\approx 5,449489743$$

Lahenduse etapid

Tegurda 18=3^{2}\times 2. Kirjutage \sqrt{3^{2}\times 2} toote juured, kui see ruut \sqrt{3^{2}}\sqrt{2}. Leidke 3^{2} ruutjuur.

$$\frac{3\sqrt{2}-\sqrt{12}}{\sqrt{50}-\sqrt{48}}$$

Tegurda 12=2^{2}\times 3. Kirjutage \sqrt{2^{2}\times 3} toote juured, kui see ruut \sqrt{2^{2}}\sqrt{3}. Leidke 2^{2} ruutjuur.

$$\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{50}-\sqrt{48}}$$

Tegurda 50=5^{2}\times 2. Kirjutage \sqrt{5^{2}\times 2} toote juured, kui see ruut \sqrt{5^{2}}\sqrt{2}. Leidke 5^{2} ruutjuur.

$$\frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-\sqrt{48}}$$

Tegurda 48=4^{2}\times 3. Kirjutage \sqrt{4^{2}\times 3} toote juured, kui see ruut \sqrt{4^{2}}\sqrt{3}. Leidke 4^{2} ruutjuur.

$$\frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-4\sqrt{3}}$$

Ratsionaliseerige korrutades lugeja ja 5\sqrt{2}+4\sqrt{3} nimetaja \frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-4\sqrt{3}} nimetaja.

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

Mõelge valemile \left(5\sqrt{2}-4\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Arvutage 2 aste 5 ja leidke 25.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{25\left(\sqrt{2}\right)^{2}-\left(-4\sqrt{3}\right)^{2}}$$

\sqrt{2} ruut on 2.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{25\times 2-\left(-4\sqrt{3}\right)^{2}}$$

Korrutage 25 ja 2, et leida 50.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-\left(-4\sqrt{3}\right)^{2}}$$

Laiendage \left(-4\sqrt{3}\right)^{2}.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Arvutage 2 aste -4 ja leidke 16.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-16\left(\sqrt{3}\right)^{2}}$$

\sqrt{3} ruut on 3.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-16\times 3}$$

Korrutage 16 ja 3, et leida 48.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-48}$$

Lahutage 48 väärtusest 50, et leida 2.

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

Rakendage distributiivsusomadus, korrutades avaldise 3\sqrt{2}-2\sqrt{3} iga liikme avaldise 5\sqrt{2}+4\sqrt{3} iga liikmega.

$$\frac{15\left(\sqrt{2}\right)^{2}+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

\sqrt{2} ruut on 2.

$$\frac{15\times 2+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

Korrutage 15 ja 2, et leida 30.

$$\frac{30+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

\sqrt{3} ja \sqrt{2} korrutage numbrid, mis on sama juur.

$$\frac{30+12\sqrt{6}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

\sqrt{3} ja \sqrt{2} korrutage numbrid, mis on sama juur.

$$\frac{30+12\sqrt{6}-10\sqrt{6}-8\left(\sqrt{3}\right)^{2}}{2}$$

Kombineerige 12\sqrt{6} ja -10\sqrt{6}, et leida 2\sqrt{6}.

$$\frac{30+2\sqrt{6}-8\left(\sqrt{3}\right)^{2}}{2}$$

\sqrt{3} ruut on 3.

$$\frac{30+2\sqrt{6}-8\times 3}{2}$$

Korrutage -8 ja 3, et leida -24.

$$\frac{30+2\sqrt{6}-24}{2}$$

Lahutage 24 väärtusest 30, et leida 6.

$$\frac{6+2\sqrt{6}}{2}$$

Jagage 6+2\sqrt{6} iga liige 2-ga, et saada 3+\sqrt{6}.

$$3+\sqrt{6}$$