Lahenda frac{{left(sqrt{2}right)}^2+{left(frac{sqrt{6}+sqrt{2}}{2}right)}^2-{left(sqrt{3}right)}^2}{2*sqrt{2}*(frac{sqrt{6}+sqrt{2}}{2})}

$$\frac{ { \left( \sqrt{ 2 } \right) }^{ 2 } + { \left( \frac{ \sqrt{ 6 } + \sqrt{ 2 } }{ 2 } \right) }^{ 2 } - { \left( \sqrt{ 3 } \right) }^{ 2 } }{ 2 \times \sqrt{ 2 } \times ( \frac{ \sqrt{ 6 } + \sqrt{ 2 } }{ 2 } ) }$$

Arvuta

$$\frac{1}{2}=0,5$$

Lühike lahenduskäik

\sqrt{2} ruut on 2.

$$\frac{2+\left(\frac{\sqrt{6}+\sqrt{2}}{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Avaldise \frac{\sqrt{6}+\sqrt{2}}{2} astendamiseks astendage nii lugeja kui ka nimetaja ning seejärel tehke jagamistehe.

$$\frac{2+\frac{\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Avaldiste liitmiseks või lahutamiseks laiendage need, et neil oleksid ühised nimetajad. Korrutage omavahel 2 ja \frac{2^{2}}{2^{2}}.

$$\frac{\frac{2\times 2^{2}}{2^{2}}+\frac{\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Kuna murdudel \frac{2\times 2^{2}}{2^{2}} ja \frac{\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}} on sama nimetaja, liitke nende lugejad.

$$\frac{\frac{2\times 2^{2}+\left(\sqrt{6}+\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Tehke korrutustehted võrrandis 2\times 2^{2}+\left(\sqrt{6}+\sqrt{2}\right)^{2}.

$$\frac{\frac{8+\left(\sqrt{6}\right)^{2}+2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Tehke arvutustehted avaldises 8+\left(\sqrt{6}\right)^{2}+2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2}.

$$\frac{\frac{16+4\sqrt{3}}{2^{2}}-\left(\sqrt{3}\right)^{2}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

\sqrt{3} ruut on 3.

$$\frac{\frac{16+4\sqrt{3}}{2^{2}}-3}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Avaldiste liitmiseks või lahutamiseks laiendage need, et neil oleksid ühised nimetajad. Korrutage omavahel 3 ja \frac{2^{2}}{2^{2}}.

$$\frac{\frac{16+4\sqrt{3}}{2^{2}}-\frac{3\times 2^{2}}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Kuna murdudel \frac{16+4\sqrt{3}}{2^{2}} ja \frac{3\times 2^{2}}{2^{2}} on sama nimetaja, lahutage nende lugejad.

$$\frac{\frac{16+4\sqrt{3}-3\times 2^{2}}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Tehke korrutustehted võrrandis 16+4\sqrt{3}-3\times 2^{2}.

$$\frac{\frac{16+4\sqrt{3}-12}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Tehke arvutustehted avaldises 16+4\sqrt{3}-12.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}}{2\sqrt{2}\times \frac{\sqrt{6}+\sqrt{2}}{2}}$$

Taandage 2 ja 2.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}}{\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}}$$

Ratsionaliseerige korrutades lugeja ja \sqrt{2} nimetaja \frac{\frac{4+4\sqrt{3}}{2^{2}}}{\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}} nimetaja.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{2}\right)^{2}}$$

\sqrt{2} ruut on 2.

$$\frac{\frac{4+4\sqrt{3}}{2^{2}}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

Arvutage 2 aste 2 ja leidke 4.

$$\frac{\frac{4+4\sqrt{3}}{4}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

Jagage 4+4\sqrt{3} iga liige 4-ga, et saada 1+\sqrt{3}.

$$\frac{\left(1+\sqrt{3}\right)\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

Kasutage distributiivsusomadust, et korrutada 1+\sqrt{3} ja \sqrt{2}.

$$\frac{\sqrt{2}+\sqrt{3}\sqrt{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

\sqrt{3} ja \sqrt{2} korrutage numbrid, mis on sama juur.

$$\frac{\sqrt{2}+\sqrt{6}}{\left(\sqrt{6}+\sqrt{2}\right)\times 2}$$

Kasutage distributiivsusomadust, et korrutada \sqrt{6}+\sqrt{2} ja 2.

$$\frac{\sqrt{2}+\sqrt{6}}{2\sqrt{6}+2\sqrt{2}}$$

Ratsionaliseerige korrutades lugeja ja 2\sqrt{6}-2\sqrt{2} nimetaja \frac{\sqrt{2}+\sqrt{6}}{2\sqrt{6}+2\sqrt{2}} nimetaja.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{\left(2\sqrt{6}+2\sqrt{2}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}$$

Mõelge valemile \left(2\sqrt{6}+2\sqrt{2}\right)\left(2\sqrt{6}-2\sqrt{2}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{\left(2\sqrt{6}\right)^{2}-\left(2\sqrt{2}\right)^{2}}$$

Laiendage \left(2\sqrt{6}\right)^{2}.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{2^{2}\left(\sqrt{6}\right)^{2}-\left(2\sqrt{2}\right)^{2}}$$

Arvutage 2 aste 2 ja leidke 4.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{4\left(\sqrt{6}\right)^{2}-\left(2\sqrt{2}\right)^{2}}$$

\sqrt{6} ruut on 6.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{4\times 6-\left(2\sqrt{2}\right)^{2}}$$

Korrutage 4 ja 6, et leida 24.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-\left(2\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-2^{2}\left(\sqrt{2}\right)^{2}}$$

Arvutage 2 aste 2 ja leidke 4.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-4\left(\sqrt{2}\right)^{2}}$$

\sqrt{2} ruut on 2.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-4\times 2}$$

Korrutage 4 ja 2, et leida 8.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{24-8}$$

Lahutage 8 väärtusest 24, et leida 16.

$$\frac{\left(\sqrt{2}+\sqrt{6}\right)\left(2\sqrt{6}-2\sqrt{2}\right)}{16}$$

Kasutage distributiivsusomadust, et korrutada \sqrt{2}+\sqrt{6} ja 2\sqrt{6}-2\sqrt{2}, ning koondage sarnased liikmed.

$$\frac{-2\left(\sqrt{2}\right)^{2}+2\left(\sqrt{6}\right)^{2}}{16}$$

\sqrt{2} ruut on 2.

$$\frac{-2\times 2+2\left(\sqrt{6}\right)^{2}}{16}$$

Korrutage -2 ja 2, et leida -4.

$$\frac{-4+2\left(\sqrt{6}\right)^{2}}{16}$$

\sqrt{6} ruut on 6.

$$\frac{-4+2\times 6}{16}$$

Korrutage 2 ja 6, et leida 12.

$$\frac{-4+12}{16}$$

Liitke -4 ja 12, et leida 8.

$$\frac{8}{16}$$

Taandage murd \frac{8}{16} vähimale ühiskordsele, eraldades ja taandades arvu 8.

$$\frac{1}{2}$$

Lahuta teguriteks

$$\frac{1}{2} = 0,5$$