Issolvi 1/2sqrt{48}div(3sqrt{2}-sqrt{3})

$$\frac { 1 } { 2 } \sqrt { 48 } \div ( 3 \sqrt { 2 } - \sqrt { 3 } )$$

Evalwa

$$\frac{2\left(\sqrt{6}+1\right)}{5}\approx 1.379795897$$

Passi tas-Soluzzjonijiet

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}.

$$\frac{\frac{1}{2}\times 4\sqrt{3}}{3\sqrt{2}-\sqrt{3}}$$

Immultiplika \frac{1}{2} u 4 biex tikseb \frac{4}{2}.

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

Iddividi 4 b'2 biex tikseb2.

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

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

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

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

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

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

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

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

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

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

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

Immultiplika 9 u 2 biex tikseb 18.

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

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

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

Naqqas 3 minn 18 biex tikseb 15.

$$\frac{2\sqrt{3}\left(3\sqrt{2}+\sqrt{3}\right)}{15}$$

Uża l-propjetà distributtiva biex timmultiplika 2\sqrt{3} b'3\sqrt{2}+\sqrt{3}.

$$\frac{6\sqrt{3}\sqrt{2}+2\left(\sqrt{3}\right)^{2}}{15}$$

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

$$\frac{6\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{15}$$

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

$$\frac{6\sqrt{6}+2\times 3}{15}$$

Immultiplika 2 u 3 biex tikseb 6.

$$\frac{6\sqrt{6}+6}{15}$$