Lutasin ang -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}}

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

I-evaluate

$$5$$

Maiikling Hakbang sa Solution

Gamitin ang binomial theorem na \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para palawakin ang \left(\sqrt{2}-1\right)^{2}.

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

Ang square ng \sqrt{2} ay 2.

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

Idagdag ang 2 at 1 para makuha ang 3.

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

I-rationalize ang denominator ng \frac{3-2\sqrt{2}}{4\sqrt{2}} sa pamamagitan ng pag-multiply ng numerator at denominator sa \sqrt{2}.

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

Ang square ng \sqrt{2} ay 2.

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

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

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

Gamitin ang binomial theorem na \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para palawakin ang \left(\sqrt{5}+\sqrt{3}\right)^{2}.

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

Ang square ng \sqrt{5} ay 5.

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

Para i-multiply ang \sqrt{5} at \sqrt{3}, i-multiply ang mga numero sa ilalim ng square root.

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

Ang square ng \sqrt{3} ay 3.

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

Idagdag ang 5 at 3 para makuha ang 8.

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

I-rationalize ang denominator ng \frac{8+2\sqrt{15}}{\sqrt{15}} sa pamamagitan ng pag-multiply ng numerator at denominator sa \sqrt{15}.

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

Ang square ng \sqrt{15} ay 15.

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

Gamitin ang binomial theorem na \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para palawakin ang \left(\sqrt{2}+1\right)^{2}.

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

Ang square ng \sqrt{2} ay 2.

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

Idagdag ang 2 at 1 para makuha ang 3.

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

I-rationalize ang denominator ng \frac{3+2\sqrt{2}}{4\sqrt{2}} sa pamamagitan ng pag-multiply ng numerator at denominator sa \sqrt{2}.

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

Ang square ng \sqrt{2} ay 2.

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

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

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

Gamitin ang binomial theorem na \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para palawakin ang \left(\sqrt{5}-\sqrt{3}\right)^{2}.

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

Ang square ng \sqrt{5} ay 5.

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

Para i-multiply ang \sqrt{5} at \sqrt{3}, i-multiply ang mga numero sa ilalim ng square root.

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

Ang square ng \sqrt{3} ay 3.

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

Idagdag ang 5 at 3 para makuha ang 8.

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

I-rationalize ang denominator ng \frac{8-2\sqrt{15}}{\sqrt{15}} sa pamamagitan ng pag-multiply ng numerator at denominator sa \sqrt{15}.

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

Ang square ng \sqrt{15} ay 15.

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

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng 8 at 15 ay 120. I-multiply ang -\frac{\left(3-2\sqrt{2}\right)\sqrt{2}}{8} times \frac{15}{15}. I-multiply ang \frac{\left(8+2\sqrt{15}\right)\sqrt{15}}{15} times \frac{8}{8}.

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

Dahil may parehong denominator ang -\frac{15\left(3-2\sqrt{2}\right)\sqrt{2}}{120} at \frac{8\left(8+2\sqrt{15}\right)\sqrt{15}}{120}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

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

Gawin ang mga pag-multiply sa -15\left(3-2\sqrt{2}\right)\sqrt{2}+8\left(8+2\sqrt{15}\right)\sqrt{15}.

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

Kalkulahin ang -45\sqrt{2}+60+64\sqrt{15}+240.

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

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng 120 at 8 ay 120. I-multiply ang \frac{\left(3+2\sqrt{2}\right)\sqrt{2}}{8} times \frac{15}{15}.

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

Dahil may parehong denominator ang \frac{-45\sqrt{2}+300+64\sqrt{15}}{120} at \frac{15\left(3+2\sqrt{2}\right)\sqrt{2}}{120}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

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

Gawin ang mga pag-multiply sa -45\sqrt{2}+300+64\sqrt{15}+15\left(3+2\sqrt{2}\right)\sqrt{2}.

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

Kalkulahin ang -45\sqrt{2}+300+64\sqrt{15}+45\sqrt{2}+60.

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

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng 120 at 15 ay 120. I-multiply ang \frac{\left(8-2\sqrt{15}\right)\sqrt{15}}{15} times \frac{8}{8}.

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

Dahil may parehong denominator ang \frac{360+64\sqrt{15}}{120} at \frac{8\left(8-2\sqrt{15}\right)\sqrt{15}}{120}, ibawas ang mga ito sa pamamagitan ng pagbawas sa mga numerator ng mga ito.

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

Gawin ang mga pag-multiply sa 360+64\sqrt{15}-8\left(8-2\sqrt{15}\right)\sqrt{15}.

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

Kalkulahin ang 360+64\sqrt{15}-64\sqrt{15}+240.

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

I-divide ang 600 gamit ang 120 para makuha ang 5.

$$5$$

I-factor

$$5$$