Lahenda 2/sqrt{8+sqrt{7}}=

$$\frac { 2 } { \sqrt { 8 } + \sqrt { 7 } } =$$

Arvuta

$$2^{\frac{5}{2}}-2\sqrt{7}\approx 0,365351627$$

Lahenduse etapid

Tegurda 8=2^{2}\times 2. Kirjutage \sqrt{2^{2}\times 2} toote juured, kui see ruut \sqrt{2^{2}}\sqrt{2}. Leidke 2^{2} ruutjuur.

$$\frac{2}{2\sqrt{2}+\sqrt{7}}$$

Ratsionaliseerige korrutades lugeja ja 2\sqrt{2}-\sqrt{7} nimetaja \frac{2}{2\sqrt{2}+\sqrt{7}} nimetaja.

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

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

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

Laiendage \left(2\sqrt{2}\right)^{2}.

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

Arvutage 2 aste 2 ja leidke 4.

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

\sqrt{2} ruut on 2.

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

Korrutage 4 ja 2, et leida 8.

$$\frac{2\left(2\sqrt{2}-\sqrt{7}\right)}{8-\left(\sqrt{7}\right)^{2}}$$

\sqrt{7} ruut on 7.

$$\frac{2\left(2\sqrt{2}-\sqrt{7}\right)}{8-7}$$

Lahutage 7 väärtusest 8, et leida 1.

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

Arv jagatuna ühega annab tulemiks arvu enda.

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

Kasutage distributiivsusomadust, et korrutada 2 ja 2\sqrt{2}-\sqrt{7}.

$$4\sqrt{2}-2\sqrt{7}$$