Lutasin ang frac{sqrt{5}div[sqrt{8}+sqrt{5}]}{sqrt{8}-sqrt{5}}

$$\frac { \sqrt { 5 } \div [ \sqrt { 8 } + \sqrt { 5 } ] } { \sqrt { 8 } - \sqrt { 5 } }$$

I-evaluate

$$\frac{\sqrt{5}}{3}\approx 0.745355992$$

Mga Hakbang ng Solution

I-factor out ang 8=2^{2}\times 2. I-rewrite ang square root ng product na \sqrt{2^{2}\times 2} bilang product ng mga square root na \sqrt{2^{2}}\sqrt{2}. Kunin ang square root ng 2^{2}.

$$\frac{\frac{\sqrt{5}}{2\sqrt{2}+\sqrt{5}}}{\sqrt{8}-\sqrt{5}}$$

I-rationalize ang denominator ng \frac{\sqrt{5}}{2\sqrt{2}+\sqrt{5}} sa pamamagitan ng pag-multiply ng numerator at denominator sa 2\sqrt{2}-\sqrt{5}.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{\left(2\sqrt{2}+\sqrt{5}\right)\left(2\sqrt{2}-\sqrt{5}\right)}}{\sqrt{8}-\sqrt{5}}$$

Isaalang-alang ang \left(2\sqrt{2}+\sqrt{5}\right)\left(2\sqrt{2}-\sqrt{5}\right). Maaaring ma-transform ang pag-multiply sa difference ng mga square gamit ang rule na: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{\left(2\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

Palawakin ang \left(2\sqrt{2}\right)^{2}.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{2^{2}\left(\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

Kalkulahin ang 2 sa power ng 2 at kunin ang 4.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{4\left(\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

Ang square ng \sqrt{2} ay 2.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{4\times 2-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

I-multiply ang 4 at 2 para makuha ang 8.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{8-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

Ang square ng \sqrt{5} ay 5.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{8-5}}{\sqrt{8}-\sqrt{5}}$$

I-subtract ang 5 mula sa 8 para makuha ang 3.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3}}{\sqrt{8}-\sqrt{5}}$$

I-factor out ang 8=2^{2}\times 2. I-rewrite ang square root ng product na \sqrt{2^{2}\times 2} bilang product ng mga square root na \sqrt{2^{2}}\sqrt{2}. Kunin ang square root ng 2^{2}.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3}}{2\sqrt{2}-\sqrt{5}}$$

Ipakita ang \frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3}}{2\sqrt{2}-\sqrt{5}} bilang isang single fraction.

$$\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3\left(2\sqrt{2}-\sqrt{5}\right)}$$

I-cancel out ang -\sqrt{5}+2\sqrt{2} sa parehong numerator at denominator.

$$\frac{\sqrt{5}}{3}$$