Issolvi frac{2(sqrt{3}+sqrt{2})}{4-sqrt{6}}*frac{9-sqrt{6}}{4-sqrt{6}}

$$\frac { 2 ( \sqrt { 3 } + \sqrt { 2 } ) } { 4 - \sqrt { 6 } } \cdot \frac { 9 - \sqrt { 6 } } { 4 - \sqrt { 6 } }$$

Evalwa

$$5\sqrt{3}+6\sqrt{2}\approx 17.145535412$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{2\left(\sqrt{3}+\sqrt{2}\right)}{4-\sqrt{6}} billi timmultiplika in-numeratur u d-denominatur mill-4+\sqrt{6}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{\left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right)}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Ikkunsidra li \left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{4^{2}-\left(\sqrt{6}\right)^{2}}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Ikkwadra 4. Ikkwadra \sqrt{6}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{16-6}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Naqqas 6 minn 16 biex tikseb 10.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{9-\sqrt{6}}{4-\sqrt{6}}$$

Irrazzjonalizza d-denominatur tal-\frac{9-\sqrt{6}}{4-\sqrt{6}} billi timmultiplika in-numeratur u d-denominatur mill-4+\sqrt{6}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{\left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right)}$$

Ikkunsidra li \left(4-\sqrt{6}\right)\left(4+\sqrt{6}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{4^{2}-\left(\sqrt{6}\right)^{2}}$$

Ikkwadra 4. Ikkwadra \sqrt{6}.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{16-6}$$

Naqqas 6 minn 16 biex tikseb 10.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10}\times \frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{10}$$

Immultiplika \frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)}{10} b'\frac{\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{10} billi timmultiplika n-numeratur bin-numeratur u d-denominatur bid-denominatur.

$$\frac{2\left(\sqrt{3}+\sqrt{2}\right)\left(4+\sqrt{6}\right)\left(9-\sqrt{6}\right)\left(4+\sqrt{6}\right)}{10\times 10}$$

Annulla 2 fin-numeratur u d-denominatur.

$$\frac{\left(\sqrt{6}+4\right)\left(\sqrt{6}+4\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Immultiplika \sqrt{6}+4 u \sqrt{6}+4 biex tikseb \left(\sqrt{6}+4\right)^{2}.

$$\frac{\left(\sqrt{6}+4\right)^{2}\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(\sqrt{6}+4\right)^{2}.

$$\frac{\left(\left(\sqrt{6}\right)^{2}+8\sqrt{6}+16\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Il-kwadrat ta' \sqrt{6} huwa 6.

$$\frac{\left(6+8\sqrt{6}+16\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Żid 6 u 16 biex tikseb 22.

$$\frac{\left(22+8\sqrt{6}\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{5\times 10}$$

Immultiplika 5 u 10 biex tikseb 50.

$$\frac{\left(22+8\sqrt{6}\right)\left(-\sqrt{6}+9\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 22+8\sqrt{6} b'kull terminu ta' -\sqrt{6}+9.

$$\frac{\left(-22\sqrt{6}+198-8\left(\sqrt{6}\right)^{2}+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Il-kwadrat ta' \sqrt{6} huwa 6.

$$\frac{\left(-22\sqrt{6}+198-8\times 6+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Immultiplika -8 u 6 biex tikseb -48.

$$\frac{\left(-22\sqrt{6}+198-48+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Naqqas 48 minn 198 biex tikseb 150.

$$\frac{\left(-22\sqrt{6}+150+72\sqrt{6}\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Ikkombina -22\sqrt{6} u 72\sqrt{6} biex tikseb 50\sqrt{6}.

$$\frac{\left(50\sqrt{6}+150\right)\left(\sqrt{2}+\sqrt{3}\right)}{50}$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 50\sqrt{6}+150 b'kull terminu ta' \sqrt{2}+\sqrt{3}.

$$\frac{50\sqrt{6}\sqrt{2}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

Iffattura 6=2\times 3. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{2\times 3} bħala l-prodott tal-għeruq kwadrati \sqrt{2}\sqrt{3}.

$$\frac{50\sqrt{2}\sqrt{3}\sqrt{2}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

Immultiplika \sqrt{2} u \sqrt{2} biex tikseb 2.

$$\frac{50\times 2\sqrt{3}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

Immultiplika 50 u 2 biex tikseb 100.

$$\frac{100\sqrt{3}+50\sqrt{6}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

Iffattura 6=3\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{3\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{3}\sqrt{2}.

$$\frac{100\sqrt{3}+50\sqrt{3}\sqrt{2}\sqrt{3}+150\sqrt{2}+150\sqrt{3}}{50}$$

Immultiplika \sqrt{3} u \sqrt{3} biex tikseb 3.

$$\frac{100\sqrt{3}+50\times 3\sqrt{2}+150\sqrt{2}+150\sqrt{3}}{50}$$

Immultiplika 50 u 3 biex tikseb 150.

$$\frac{100\sqrt{3}+150\sqrt{2}+150\sqrt{2}+150\sqrt{3}}{50}$$

Ikkombina 150\sqrt{2} u 150\sqrt{2} biex tikseb 300\sqrt{2}.

$$\frac{100\sqrt{3}+300\sqrt{2}+150\sqrt{3}}{50}$$

Ikkombina 100\sqrt{3} u 150\sqrt{3} biex tikseb 250\sqrt{3}.

$$\frac{250\sqrt{3}+300\sqrt{2}}{50}$$

Iddividi kull terminu ta' 250\sqrt{3}+300\sqrt{2} b'50 biex tikseb5\sqrt{3}+6\sqrt{2}.

$$5\sqrt{3}+6\sqrt{2}$$

Fattur

$$5 \sqrt{3} + 6 \sqrt{2} = 17.145535412$$