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

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

Evalwa

$$\frac{\sqrt{502}+5\sqrt{2}}{904}\approx 0.032606664$$

Passi tas-Soluzzjonijiet

Iffattura 2008=2^{2}\times 502. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{2^{2}\times 502} bħala l-prodott tal-għeruq kwadrati \sqrt{2^{2}}\sqrt{502}. Ħu l-għerq kwadrat ta' 2^{2}.

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

Iffattura 200=10^{2}\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{10^{2}\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{10^{2}}\sqrt{2}. Ħu l-għerq kwadrat ta' 10^{2}.

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

Irrazzjonalizza d-denominatur tal-\frac{1}{2\sqrt{502}-10\sqrt{2}} billi timmultiplika in-numeratur u d-denominatur mill-2\sqrt{502}+10\sqrt{2}.

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

Ikkunsidra li \left(2\sqrt{502}-10\sqrt{2}\right)\left(2\sqrt{502}+10\sqrt{2}\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}.

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

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

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

Ikkalkula 2 bil-power ta' 2 u tikseb 4.

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

Il-kwadrat ta' \sqrt{502} huwa 502.

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

Immultiplika 4 u 502 biex tikseb 2008.

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

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

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

Ikkalkula -10 bil-power ta' 2 u tikseb 100.

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

Il-kwadrat ta' \sqrt{2} huwa 2.

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

Immultiplika 100 u 2 biex tikseb 200.

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

Naqqas 200 minn 2008 biex tikseb 1808.

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