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

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

Evalwa

$$\sqrt{6}+3\approx 5.449489743$$

Passi tas-Soluzzjonijiet

Iffattura 18=3^{2}\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{3^{2}\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{3^{2}}\sqrt{2}. Ħu l-għerq kwadrat ta' 3^{2}.

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

Iffattura 12=2^{2}\times 3. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{2^{2}\times 3} bħala l-prodott tal-għeruq kwadrati \sqrt{2^{2}}\sqrt{3}. Ħu l-għerq kwadrat ta' 2^{2}.

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

Iffattura 50=5^{2}\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{5^{2}\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{5^{2}}\sqrt{2}. Ħu l-għerq kwadrat ta' 5^{2}.

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

Iffattura 48=4^{2}\times 3. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{4^{2}\times 3} bħala l-prodott tal-għeruq kwadrati \sqrt{4^{2}}\sqrt{3}. Ħu l-għerq kwadrat ta' 4^{2}.

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

Irrazzjonalizza d-denominatur tal-\frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-4\sqrt{3}} billi timmultiplika in-numeratur u d-denominatur mill-5\sqrt{2}+4\sqrt{3}.

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

Ikkunsidra li \left(5\sqrt{2}-4\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\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}.

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

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

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

Ikkalkula 5 bil-power ta' 2 u tikseb 25.

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

Il-kwadrat ta' \sqrt{2} huwa 2.

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

Immultiplika 25 u 2 biex tikseb 50.

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

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

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

Ikkalkula -4 bil-power ta' 2 u tikseb 16.

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

Il-kwadrat ta' \sqrt{3} huwa 3.

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

Immultiplika 16 u 3 biex tikseb 48.

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

Naqqas 48 minn 50 biex tikseb 2.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 3\sqrt{2}-2\sqrt{3} b'kull terminu ta' 5\sqrt{2}+4\sqrt{3}.

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

Il-kwadrat ta' \sqrt{2} huwa 2.

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

Immultiplika 15 u 2 biex tikseb 30.

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

Biex timmultiplika \sqrt{3} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

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

Biex timmultiplika \sqrt{3} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

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

Ikkombina 12\sqrt{6} u -10\sqrt{6} biex tikseb 2\sqrt{6}.

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

Il-kwadrat ta' \sqrt{3} huwa 3.

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

Immultiplika -8 u 3 biex tikseb -24.

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

Naqqas 24 minn 30 biex tikseb 6.

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

Iddividi kull terminu ta' 6+2\sqrt{6} b'2 biex tikseb3+\sqrt{6}.

$$3+\sqrt{6}$$