Issolvi frac{2+sqrt{3}}{3sqrt{2}+2sqrt{7}}

$$\frac { 2 + \sqrt { 3 } } { 3 \sqrt { 2 } + 2 \sqrt { 7 } }$$

Evalwa

$$\frac{\sqrt{21}}{5}+\frac{2\sqrt{7}}{5}-\frac{3\sqrt{2}}{5}-\frac{3\sqrt{6}}{10}\approx 0.391440603$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{2+\sqrt{3}}{3\sqrt{2}+2\sqrt{7}} billi timmultiplika in-numeratur u d-denominatur mill-3\sqrt{2}-2\sqrt{7}.

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{\left(3\sqrt{2}+2\sqrt{7}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}$$

Ikkunsidra li \left(3\sqrt{2}+2\sqrt{7}\right)\left(3\sqrt{2}-2\sqrt{7}\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}.

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{\left(3\sqrt{2}\right)^{2}-\left(2\sqrt{7}\right)^{2}}$$

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

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(2\sqrt{7}\right)^{2}}$$

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

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{9\left(\sqrt{2}\right)^{2}-\left(2\sqrt{7}\right)^{2}}$$

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

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{9\times 2-\left(2\sqrt{7}\right)^{2}}$$

Immultiplika 9 u 2 biex tikseb 18.

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{18-\left(2\sqrt{7}\right)^{2}}$$

Espandi \left(2\sqrt{7}\right)^{2}.

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{18-2^{2}\left(\sqrt{7}\right)^{2}}$$

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

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{18-4\left(\sqrt{7}\right)^{2}}$$

Il-kwadrat ta' \sqrt{7} huwa 7.

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{18-4\times 7}$$

Immultiplika 4 u 7 biex tikseb 28.

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{18-28}$$

Naqqas 28 minn 18 biex tikseb -10.

$$\frac{\left(2+\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{7}\right)}{-10}$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 2+\sqrt{3} b'kull terminu ta' 3\sqrt{2}-2\sqrt{7}.

$$\frac{6\sqrt{2}-4\sqrt{7}+3\sqrt{3}\sqrt{2}-2\sqrt{3}\sqrt{7}}{-10}$$

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

$$\frac{6\sqrt{2}-4\sqrt{7}+3\sqrt{6}-2\sqrt{3}\sqrt{7}}{-10}$$

Biex timmultiplika \sqrt{3} u \sqrt{7}, immultiplika n-numri taħt l-għerq kwadrat.

$$\frac{6\sqrt{2}-4\sqrt{7}+3\sqrt{6}-2\sqrt{21}}{-10}$$

Immultiplika kemm in-numeratur u kif ukoll id-denominatur b’-1.

$$\frac{-6\sqrt{2}+4\sqrt{7}-3\sqrt{6}+2\sqrt{21}}{10}$$