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

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

Arvuta

$$\frac{28\sqrt{6}}{43}\approx 1,595016577$$

Lühike lahenduskäik

Ratsionaliseerige korrutades lugeja ja 7+\sqrt{6} nimetaja \frac{7+\sqrt{6}}{7-\sqrt{6}} nimetaja.

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

Mõelge valemile \left(7-\sqrt{6}\right)\left(7+\sqrt{6}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Tõstke 7 ruutu. Tõstke \sqrt{6} ruutu.

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

Lahutage 6 väärtusest 49, et leida 43.

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

Korrutage 7+\sqrt{6} ja 7+\sqrt{6}, et leida \left(7+\sqrt{6}\right)^{2}.

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

Kasutage kaksliikme \left(7+\sqrt{6}\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

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

\sqrt{6} ruut on 6.

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

Liitke 49 ja 6, et leida 55.

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

Ratsionaliseerige korrutades lugeja ja 7-\sqrt{6} nimetaja \frac{7-\sqrt{6}}{7+\sqrt{6}} nimetaja.

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

Mõelge valemile \left(7+\sqrt{6}\right)\left(7-\sqrt{6}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Tõstke 7 ruutu. Tõstke \sqrt{6} ruutu.

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

Lahutage 6 väärtusest 49, et leida 43.

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

Korrutage 7-\sqrt{6} ja 7-\sqrt{6}, et leida \left(7-\sqrt{6}\right)^{2}.

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

Kasutage kaksliikme \left(7-\sqrt{6}\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

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

\sqrt{6} ruut on 6.

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

Liitke 49 ja 6, et leida 55.

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

Kuna murdudel \frac{55+14\sqrt{6}}{43} ja \frac{55-14\sqrt{6}}{43} on sama nimetaja, lahutage nende lugejad.

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

Tehke korrutustehted võrrandis 55+14\sqrt{6}-\left(55-14\sqrt{6}\right).

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

Tehke arvutustehted avaldises 55+14\sqrt{6}-55+14\sqrt{6}.

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