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

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

Evalwa

$$4\sqrt{3}+2-2\sqrt{6}\approx 4.029223745$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(2+\sqrt{3}\right)^{2}.

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

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

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

Żid 4 u 3 biex tikseb 7.

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

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

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

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

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

Ikkwadra \sqrt{3}. Ikkwadra \sqrt{2}.

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

Naqqas 2 minn 3 biex tikseb 1.

$$7+4\sqrt{3}-\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}{1}$$

Kwalunkwe ħaġa diviża b'wieħed tagħti riżultat tagħha stess.

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

Immultiplika \sqrt{3}+\sqrt{2} u \sqrt{3}+\sqrt{2} biex tikseb \left(\sqrt{3}+\sqrt{2}\right)^{2}.

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

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(\sqrt{3}+\sqrt{2}\right)^{2}.

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

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

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

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

$$7+4\sqrt{3}-\left(3+2\sqrt{6}+\left(\sqrt{2}\right)^{2}\right)$$

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

$$7+4\sqrt{3}-\left(3+2\sqrt{6}+2\right)$$

Żid 3 u 2 biex tikseb 5.

$$7+4\sqrt{3}-\left(5+2\sqrt{6}\right)$$

Biex issib l-oppost ta' 5+2\sqrt{6}, sib l-oppost ta' kull terminu.

$$7+4\sqrt{3}-5-2\sqrt{6}$$

Naqqas 5 minn 7 biex tikseb 2.

$$2+4\sqrt{3}-2\sqrt{6}$$