Lahenda -frac{(sqrt{2}-1)^2}{4sqrt{2}}+frac{(sqrt{5}+sqrt{3})^2}{sqrt{15}}+frac{(sqrt{2}+1)^2}{4sqrt{2}}-frac{(sqrt{5}-sqrt{3})^2}{sqrt{15}}

$$- \frac { ( \sqrt { 2 } - 1 ) ^ { 2 } } { 4 \sqrt { 2 } } + \frac { ( \sqrt { 5 } + \sqrt { 3 } ) ^ { 2 } } { \sqrt { 15 } } + \frac { ( \sqrt { 2 } + 1 ) ^ { 2 } } { 4 \sqrt { 2 } } - \frac { ( \sqrt { 5 } - \sqrt { 3 } ) ^ { 2 } } { \sqrt { 15 } }$$

Arvuta

$$5$$

Lühike lahenduskäik

Kasutage kaksliikme \left(\sqrt{2}-1\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

$$-\frac{\left(\sqrt{2}\right)^{2}-2\sqrt{2}+1}{4\sqrt{2}}+\frac{\left(\sqrt{5}+\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{2} ruut on 2.

$$-\frac{2-2\sqrt{2}+1}{4\sqrt{2}}+\frac{\left(\sqrt{5}+\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Liitke 2 ja 1, et leida 3.

$$-\frac{3-2\sqrt{2}}{4\sqrt{2}}+\frac{\left(\sqrt{5}+\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Ratsionaliseerige korrutades lugeja ja \sqrt{2} nimetaja \frac{3-2\sqrt{2}}{4\sqrt{2}} nimetaja.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}+\frac{\left(\sqrt{5}+\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{2} ruut on 2.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{4\times 2}+\frac{\left(\sqrt{5}+\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Korrutage 4 ja 2, et leida 8.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(\sqrt{5}+\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Kasutage kaksliikme \left(\sqrt{5}+\sqrt{3}\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(\sqrt{5}\right)^{2}+2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{5} ruut on 5.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{5+2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{5} ja \sqrt{3} korrutage numbrid, mis on sama juur.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{5+2\sqrt{15}+\left(\sqrt{3}\right)^{2}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{3} ruut on 3.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{5+2\sqrt{15}+3}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Liitke 5 ja 3, et leida 8.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{8+2\sqrt{15}}{\sqrt{15}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Ratsionaliseerige korrutades lugeja ja \sqrt{15} nimetaja \frac{8+2\sqrt{15}}{\sqrt{15}} nimetaja.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{\left(\sqrt{15}\right)^{2}}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{15} ruut on 15.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(\sqrt{2}+1\right)^{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Kasutage kaksliikme \left(\sqrt{2}+1\right)^{2} arendamiseks binoomvalemit \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(\sqrt{2}\right)^{2}+2\sqrt{2}+1}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{2} ruut on 2.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{2+2\sqrt{2}+1}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Liitke 2 ja 1, et leida 3.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{3+2\sqrt{2}}{4\sqrt{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Ratsionaliseerige korrutades lugeja ja \sqrt{2} nimetaja \frac{3+2\sqrt{2}}{4\sqrt{2}} nimetaja.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{2} ruut on 2.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{4\times 2}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Korrutage 4 ja 2, et leida 8.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{\sqrt{15}}$$

Kasutage kaksliikme \left(\sqrt{5}-\sqrt{3}\right)^{2} arendamiseks binoomvalemit \left(a-b\right)^{2}=a^{2}-2ab+b^{2}.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{5} ruut on 5.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{5-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{5} ja \sqrt{3} korrutage numbrid, mis on sama juur.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{5-2\sqrt{15}+\left(\sqrt{3}\right)^{2}}{\sqrt{15}}$$

\sqrt{3} ruut on 3.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{5-2\sqrt{15}+3}{\sqrt{15}}$$

Liitke 5 ja 3, et leida 8.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{8-2\sqrt{15}}{\sqrt{15}}$$

Ratsionaliseerige korrutades lugeja ja \sqrt{15} nimetaja \frac{8-2\sqrt{15}}{\sqrt{15}} nimetaja.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{\left(\sqrt{15}\right)^{2}}$$

\sqrt{15} ruut on 15.

$$-\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8}+\frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Avaldiste liitmiseks või lahutamiseks laiendage need, et neil oleksid ühised nimetajad. 8 ja 15 vähim ühiskordne on 120. Korrutage omavahel -\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8} ja \frac{15}{15}. Korrutage omavahel \frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15} ja \frac{8}{8}.

$$-\frac{15\left(3-2\sqrt{2}\right)\sqrt{2}}{120}+\frac{8\left(8+2\sqrt{15}\right)\sqrt{15}}{120}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Kuna murdudel -\frac{15\left(3-2\sqrt{2}\right)\sqrt{2}}{120} ja \frac{8\left(8+2\sqrt{15}\right)\sqrt{15}}{120} on sama nimetaja, liitke nende lugejad.

$$\frac{-15\left(3-2\sqrt{2}\right)\sqrt{2}+8\left(8+2\sqrt{15}\right)\sqrt{15}}{120}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Tehke korrutustehted võrrandis -15\left(3-2\sqrt{2}\right)\sqrt{2}+8\left(8+2\sqrt{15}\right)\sqrt{15}.

$$\frac{-45\sqrt{2}+60+64\sqrt{15}+240}{120}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Tehke arvutustehted avaldises -45\sqrt{2}+60+64\sqrt{15}+240.

$$\frac{-45\sqrt{2}+300+64\sqrt{15}}{120}+\frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Avaldiste liitmiseks või lahutamiseks laiendage need, et neil oleksid ühised nimetajad. 120 ja 8 vähim ühiskordne on 120. Korrutage omavahel \frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8} ja \frac{15}{15}.

$$\frac{-45\sqrt{2}+300+64\sqrt{15}}{120}+\frac{15\left(3+2\sqrt{2}\right)\sqrt{2}}{120}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Kuna murdudel \frac{-45\sqrt{2}+300+64\sqrt{15}}{120} ja \frac{15\left(3+2\sqrt{2}\right)\sqrt{2}}{120} on sama nimetaja, liitke nende lugejad.

$$\frac{-45\sqrt{2}+300+64\sqrt{15}+15\left(3+2\sqrt{2}\right)\sqrt{2}}{120}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Tehke korrutustehted võrrandis -45\sqrt{2}+300+64\sqrt{15}+15\left(3+2\sqrt{2}\right)\sqrt{2}.

$$\frac{-45\sqrt{2}+300+64\sqrt{15}+45\sqrt{2}+60}{120}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Tehke arvutustehted avaldises -45\sqrt{2}+300+64\sqrt{15}+45\sqrt{2}+60.

$$\frac{360+64\sqrt{15}}{120}-\frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15}$$

Avaldiste liitmiseks või lahutamiseks laiendage need, et neil oleksid ühised nimetajad. 120 ja 15 vähim ühiskordne on 120. Korrutage omavahel \frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15} ja \frac{8}{8}.

$$\frac{360+64\sqrt{15}}{120}-\frac{8\left(8-2\sqrt{15}\right)\sqrt{15}}{120}$$

Kuna murdudel \frac{360+64\sqrt{15}}{120} ja \frac{8\left(8-2\sqrt{15}\right)\sqrt{15}}{120} on sama nimetaja, lahutage nende lugejad.

$$\frac{360+64\sqrt{15}-8\left(8-2\sqrt{15}\right)\sqrt{15}}{120}$$

Tehke korrutustehted võrrandis 360+64\sqrt{15}-8\left(8-2\sqrt{15}\right)\sqrt{15}.

$$\frac{360+64\sqrt{15}-64\sqrt{15}+240}{120}$$

Tehke arvutustehted avaldises 360+64\sqrt{15}-64\sqrt{15}+240.

$$\frac{600}{120}$$

Jagage 600 väärtusega 120, et leida 5.

$$5$$

Lahuta teguriteks

$$5$$