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

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

Evalwa

$$\frac{47\sqrt{5}-56\sqrt{2}}{37}\approx 0.699979336$$

Passi tas-Soluzzjonijiet

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

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

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

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

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

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

Ikkalkula 3 bil-power ta' 2 u tikseb 9.

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

Il-kwadrat ta' \sqrt{5} huwa 5.

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

Immultiplika 9 u 5 biex tikseb 45.

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

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

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

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

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

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

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

Immultiplika 4 u 2 biex tikseb 8.

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

Naqqas 8 minn 45 biex tikseb 37.

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

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

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

Il-kwadrat ta' \sqrt{5} huwa 5.

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

Immultiplika 3 u 5 biex tikseb 15.

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

Biex timmultiplika \sqrt{5} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

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

Biex timmultiplika \sqrt{2} u \sqrt{5}, immultiplika n-numri taħt l-għerq kwadrat.

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

Ikkombina \sqrt{10} u -3\sqrt{10} biex tikseb -2\sqrt{10}.

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

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

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

Naqqas 2 minn 15 biex tikseb 13.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 13-2\sqrt{10} b'kull terminu ta' 3\sqrt{5}-2\sqrt{2}.

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

Iffattura 10=5\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{5\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{5}\sqrt{2}.

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

Immultiplika \sqrt{5} u \sqrt{5} biex tikseb 5.

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

Immultiplika -6 u 5 biex tikseb -30.

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

Ikkombina -26\sqrt{2} u -30\sqrt{2} biex tikseb -56\sqrt{2}.

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

Iffattura 10=2\times 5. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{2\times 5} bħala l-prodott tal-għeruq kwadrati \sqrt{2}\sqrt{5}.

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

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

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

Immultiplika 4 u 2 biex tikseb 8.

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

Ikkombina 39\sqrt{5} u 8\sqrt{5} biex tikseb 47\sqrt{5}.

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