Lahenda 1/sqrt{2008-sqrt{200}}

$$\frac { 1 } { \sqrt { 2008 } - \sqrt { 200 } }$$

Arvuta

$$\frac{\sqrt{502}+5\sqrt{2}}{904}\approx 0,032606664$$

Lahenduse etapid

Tegurda 2008=2^{2}\times 502. Kirjutage \sqrt{2^{2}\times 502} toote juured, kui see ruut \sqrt{2^{2}}\sqrt{502}. Leidke 2^{2} ruutjuur.

$$\frac{1}{2\sqrt{502}-\sqrt{200}}$$

Tegurda 200=10^{2}\times 2. Kirjutage \sqrt{10^{2}\times 2} toote juured, kui see ruut \sqrt{10^{2}}\sqrt{2}. Leidke 10^{2} ruutjuur.

$$\frac{1}{2\sqrt{502}-10\sqrt{2}}$$

Ratsionaliseerige korrutades lugeja ja 2\sqrt{502}+10\sqrt{2} nimetaja \frac{1}{2\sqrt{502}-10\sqrt{2}} nimetaja.

$$\frac{2\sqrt{502}+10\sqrt{2}}{\left(2\sqrt{502}-10\sqrt{2}\right)\left(2\sqrt{502}+10\sqrt{2}\right)}$$

Mõelge valemile \left(2\sqrt{502}-10\sqrt{2}\right)\left(2\sqrt{502}+10\sqrt{2}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{502}+10\sqrt{2}}{\left(2\sqrt{502}\right)^{2}-\left(-10\sqrt{2}\right)^{2}}$$

Laiendage \left(2\sqrt{502}\right)^{2}.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2^{2}\left(\sqrt{502}\right)^{2}-\left(-10\sqrt{2}\right)^{2}}$$

Arvutage 2 aste 2 ja leidke 4.

$$\frac{2\sqrt{502}+10\sqrt{2}}{4\left(\sqrt{502}\right)^{2}-\left(-10\sqrt{2}\right)^{2}}$$

\sqrt{502} ruut on 502.

$$\frac{2\sqrt{502}+10\sqrt{2}}{4\times 502-\left(-10\sqrt{2}\right)^{2}}$$

Korrutage 4 ja 502, et leida 2008.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-\left(-10\sqrt{2}\right)^{2}}$$

Laiendage \left(-10\sqrt{2}\right)^{2}.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-\left(-10\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Arvutage 2 aste -10 ja leidke 100.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-100\left(\sqrt{2}\right)^{2}}$$

\sqrt{2} ruut on 2.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-100\times 2}$$

Korrutage 100 ja 2, et leida 200.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-200}$$

Lahutage 200 väärtusest 2008, et leida 1808.

$$\frac{2\sqrt{502}+10\sqrt{2}}{1808}$$