Issolvi frac{7+sqrt{6}}{7-sqrt{6}}-frac{7-sqrt{6}}{7+sqrt{6}}

$$\frac { 7 + \sqrt { 6 } } { 7 - \sqrt { 6 } } - \frac { 7 - \sqrt { 6 } } { 7 + \sqrt { 6 } }$$

Evalwa

$$\frac{28\sqrt{6}}{43}\approx 1.595016577$$

Passi Qosra tas-Soluzzjonijiet

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

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

Ikkunsidra li \left(7-\sqrt{6}\right)\left(7+\sqrt{6}\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}.

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

Ikkwadra 7. Ikkwadra \sqrt{6}.

$$\frac{\left(7+\sqrt{6}\right)\left(7+\sqrt{6}\right)}{49-6}-\frac{7-\sqrt{6}}{7+\sqrt{6}}$$

Naqqas 6 minn 49 biex tikseb 43.

$$\frac{\left(7+\sqrt{6}\right)\left(7+\sqrt{6}\right)}{43}-\frac{7-\sqrt{6}}{7+\sqrt{6}}$$

Immultiplika 7+\sqrt{6} u 7+\sqrt{6} biex tikseb \left(7+\sqrt{6}\right)^{2}.

$$\frac{\left(7+\sqrt{6}\right)^{2}}{43}-\frac{7-\sqrt{6}}{7+\sqrt{6}}$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(7+\sqrt{6}\right)^{2}.

$$\frac{49+14\sqrt{6}+\left(\sqrt{6}\right)^{2}}{43}-\frac{7-\sqrt{6}}{7+\sqrt{6}}$$

Il-kwadrat ta' \sqrt{6} huwa 6.

$$\frac{49+14\sqrt{6}+6}{43}-\frac{7-\sqrt{6}}{7+\sqrt{6}}$$

Żid 49 u 6 biex tikseb 55.

$$\frac{55+14\sqrt{6}}{43}-\frac{7-\sqrt{6}}{7+\sqrt{6}}$$

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

$$\frac{55+14\sqrt{6}}{43}-\frac{\left(7-\sqrt{6}\right)\left(7-\sqrt{6}\right)}{\left(7+\sqrt{6}\right)\left(7-\sqrt{6}\right)}$$

Ikkunsidra li \left(7+\sqrt{6}\right)\left(7-\sqrt{6}\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}.

$$\frac{55+14\sqrt{6}}{43}-\frac{\left(7-\sqrt{6}\right)\left(7-\sqrt{6}\right)}{7^{2}-\left(\sqrt{6}\right)^{2}}$$

Ikkwadra 7. Ikkwadra \sqrt{6}.

$$\frac{55+14\sqrt{6}}{43}-\frac{\left(7-\sqrt{6}\right)\left(7-\sqrt{6}\right)}{49-6}$$

Naqqas 6 minn 49 biex tikseb 43.

$$\frac{55+14\sqrt{6}}{43}-\frac{\left(7-\sqrt{6}\right)\left(7-\sqrt{6}\right)}{43}$$

Immultiplika 7-\sqrt{6} u 7-\sqrt{6} biex tikseb \left(7-\sqrt{6}\right)^{2}.

$$\frac{55+14\sqrt{6}}{43}-\frac{\left(7-\sqrt{6}\right)^{2}}{43}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(7-\sqrt{6}\right)^{2}.

$$\frac{55+14\sqrt{6}}{43}-\frac{49-14\sqrt{6}+\left(\sqrt{6}\right)^{2}}{43}$$

Il-kwadrat ta' \sqrt{6} huwa 6.

$$\frac{55+14\sqrt{6}}{43}-\frac{49-14\sqrt{6}+6}{43}$$

Żid 49 u 6 biex tikseb 55.

$$\frac{55+14\sqrt{6}}{43}-\frac{55-14\sqrt{6}}{43}$$

Billi \frac{55+14\sqrt{6}}{43} u \frac{55-14\sqrt{6}}{43} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{55+14\sqrt{6}-\left(55-14\sqrt{6}\right)}{43}$$

Agħmel il-multiplikazzjonijiet fi 55+14\sqrt{6}-\left(55-14\sqrt{6}\right).

$$\frac{55+14\sqrt{6}-55+14\sqrt{6}}{43}$$

Agħmel il-kalkoli fi 55+14\sqrt{6}-55+14\sqrt{6}.

$$\frac{28\sqrt{6}}{43}$$