Issolvi frac{sqrt{7}-3sqrt{6}}{sqrt{2}-3sqrt{7}}

$$\frac { \sqrt { 7 } - 3 \sqrt { 6 } } { \sqrt { 2 } - 3 \sqrt { 7 } }$$

Evalwa

$$\frac{6\sqrt{3}+9\sqrt{42}-\sqrt{14}-21}{61}\approx 0.72093957$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{7}-3\sqrt{6}}{\sqrt{2}-3\sqrt{7}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{2}+3\sqrt{7}.

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

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

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

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

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

Espandi \left(-3\sqrt{7}\right)^{2}.

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

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

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{2-9\left(\sqrt{7}\right)^{2}}$$

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

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{2-9\times 7}$$

Immultiplika 9 u 7 biex tikseb 63.

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{2-63}$$

Naqqas 63 minn 2 biex tikseb -61.

$$\frac{\left(\sqrt{7}-3\sqrt{6}\right)\left(\sqrt{2}+3\sqrt{7}\right)}{-61}$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' \sqrt{7}-3\sqrt{6} b'kull terminu ta' \sqrt{2}+3\sqrt{7}.

$$\frac{\sqrt{7}\sqrt{2}+3\left(\sqrt{7}\right)^{2}-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

Biex timmultiplika \sqrt{7} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

$$\frac{\sqrt{14}+3\left(\sqrt{7}\right)^{2}-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

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

$$\frac{\sqrt{14}+3\times 7-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

Immultiplika 3 u 7 biex tikseb 21.

$$\frac{\sqrt{14}+21-3\sqrt{6}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

Iffattura 6=2\times 3. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{2\times 3} bħala l-prodott tal-għeruq kwadrati \sqrt{2}\sqrt{3}.

$$\frac{\sqrt{14}+21-3\sqrt{2}\sqrt{3}\sqrt{2}-9\sqrt{6}\sqrt{7}}{-61}$$

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

$$\frac{\sqrt{14}+21-3\times 2\sqrt{3}-9\sqrt{6}\sqrt{7}}{-61}$$

Immultiplika -3 u 2 biex tikseb -6.

$$\frac{\sqrt{14}+21-6\sqrt{3}-9\sqrt{6}\sqrt{7}}{-61}$$

Biex timmultiplika \sqrt{6} u \sqrt{7}, immultiplika n-numri taħt l-għerq kwadrat.

$$\frac{\sqrt{14}+21-6\sqrt{3}-9\sqrt{42}}{-61}$$

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

$$\frac{-\sqrt{14}-21+6\sqrt{3}+9\sqrt{42}}{61}$$