Lahenda frac{4-sqrt{2}}{4+sqrt{2}}-frac{4+sqrt{2}}{4-sqrt{2}}

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

Arvuta

$$-\frac{8\sqrt{2}}{7}\approx -1,616244071$$

Lühike lahenduskäik

Ratsionaliseerige korrutades lugeja ja 4-\sqrt{2} nimetaja \frac{4-\sqrt{2}}{4+\sqrt{2}} nimetaja.

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

Mõelge valemile \left(4+\sqrt{2}\right)\left(4-\sqrt{2}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Tõstke 4 ruutu. Tõstke \sqrt{2} ruutu.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{16-2}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Lahutage 2 väärtusest 16, et leida 14.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Korrutage 4-\sqrt{2} ja 4-\sqrt{2}, et leida \left(4-\sqrt{2}\right)^{2}.

$$\frac{\left(4-\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Kasutage kaksliikme \left(4-\sqrt{2}\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

$$\frac{16-8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

\sqrt{2} ruut on 2.

$$\frac{16-8\sqrt{2}+2}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Liitke 16 ja 2, et leida 18.

$$\frac{18-8\sqrt{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Ratsionaliseerige korrutades lugeja ja 4+\sqrt{2} nimetaja \frac{4+\sqrt{2}}{4-\sqrt{2}} nimetaja.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{\left(4-\sqrt{2}\right)\left(4+\sqrt{2}\right)}$$

Mõelge valemile \left(4-\sqrt{2}\right)\left(4+\sqrt{2}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{4^{2}-\left(\sqrt{2}\right)^{2}}$$

Tõstke 4 ruutu. Tõstke \sqrt{2} ruutu.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{16-2}$$

Lahutage 2 väärtusest 16, et leida 14.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{14}$$

Korrutage 4+\sqrt{2} ja 4+\sqrt{2}, et leida \left(4+\sqrt{2}\right)^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)^{2}}{14}$$

Kasutage kaksliikme \left(4+\sqrt{2}\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}$$

\sqrt{2} ruut on 2.

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+2}{14}$$

Liitke 16 ja 2, et leida 18.

$$\frac{18-8\sqrt{2}}{14}-\frac{18+8\sqrt{2}}{14}$$

Kuna murdudel \frac{18-8\sqrt{2}}{14} ja \frac{18+8\sqrt{2}}{14} on sama nimetaja, lahutage nende lugejad.

$$\frac{18-8\sqrt{2}-\left(18+8\sqrt{2}\right)}{14}$$

Tehke korrutustehted võrrandis 18-8\sqrt{2}-\left(18+8\sqrt{2}\right).

$$\frac{18-8\sqrt{2}-18-8\sqrt{2}}{14}$$

Tehke arvutustehted avaldises 18-8\sqrt{2}-18-8\sqrt{2}.

$$\frac{-16\sqrt{2}}{14}$$

Jagage -16\sqrt{2} väärtusega 14, et leida -\frac{8}{7}\sqrt{2}.

$$-\frac{8}{7}\sqrt{2}$$