Resolver frac{sqrt{5}div[sqrt{8}+sqrt{5}]}{sqrt{8}-sqrt{5}}

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

Ebaluatu

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

Ebazpidea

8=2^{2}\times 2 faktorea. Berridatzi biderketaren erro karratua (\sqrt{2^{2}\times 2}) \sqrt{2^{2}}\sqrt{2} erro karratuen biderkadura gisa. Atera 2^{2} balioaren erro karratua.

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

Adierazi \frac{\sqrt{5}}{2\sqrt{2}+\sqrt{5}} balioaren izendatzailea zenbaki arrazional gisa. Horretarako, egin zenbakitzailea eta izendatzailea bider 2\sqrt{2}-\sqrt{5}.

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

Kasurako: \left(2\sqrt{2}+\sqrt{5}\right)\left(2\sqrt{2}-\sqrt{5}\right). Biderketa karratuen desberdintasun bihur daiteke arau hau erabilita: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

4 lortzeko, egin 2 ber 2.

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

\sqrt{2} zenbakiaren karratua 2 da.

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

8 lortzeko, biderkatu 4 eta 2.

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

\sqrt{5} zenbakiaren karratua 5 da.

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

3 lortzeko, 8 balioari kendu 5.

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

8=2^{2}\times 2 faktorea. Berridatzi biderketaren erro karratua (\sqrt{2^{2}\times 2}) \sqrt{2^{2}}\sqrt{2} erro karratuen biderkadura gisa. Atera 2^{2} balioaren erro karratua.

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

Adierazi \frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3}}{2\sqrt{2}-\sqrt{5}} frakzio bakar gisa.

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

Sinplifikatu -\sqrt{5}+2\sqrt{2} zenbakitzailean eta izendatzailean.

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