Issolvi =frac{sqrt{5}+sqrt{3}}{sqrt{5}-sqrt{3}}+frac{sqrt{5}-sqrt{3}}{sqrt{5}+sqrt{3}}

$$= \frac { \sqrt { 5 } + \sqrt { 3 } } { \sqrt { 5 } - \sqrt { 3 } } + \frac { \sqrt { 5 } - \sqrt { 3 } } { \sqrt { 5 } + \sqrt { 3 } }$$

Evalwa

$$8$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{5}+\sqrt{3}.

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

Ikkunsidra li \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Ikkwadra \sqrt{5}. Ikkwadra \sqrt{3}.

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

Naqqas 3 minn 5 biex tikseb 2.

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

Immultiplika \sqrt{5}+\sqrt{3} u \sqrt{5}+\sqrt{3} biex tikseb \left(\sqrt{5}+\sqrt{3}\right)^{2}.

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

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(\sqrt{5}+\sqrt{3}\right)^{2}.

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

Il-kwadrat ta' \sqrt{5} huwa 5.

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

Biex timmultiplika \sqrt{5} u \sqrt{3}, immultiplika n-numri taħt l-għerq kwadrat.

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

Il-kwadrat ta' \sqrt{3} huwa 3.

$$\frac{5+2\sqrt{15}+3}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

Żid 5 u 3 biex tikseb 8.

$$\frac{8+2\sqrt{15}}{2}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

Iddividi kull terminu ta' 8+2\sqrt{15} b'2 biex tikseb4+\sqrt{15}.

$$4+\sqrt{15}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$$

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{5}-\sqrt{3}.

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

Ikkunsidra li \left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Ikkwadra \sqrt{5}. Ikkwadra \sqrt{3}.

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

Naqqas 3 minn 5 biex tikseb 2.

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

Immultiplika \sqrt{5}-\sqrt{3} u \sqrt{5}-\sqrt{3} biex tikseb \left(\sqrt{5}-\sqrt{3}\right)^{2}.

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

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(\sqrt{5}-\sqrt{3}\right)^{2}.

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

Il-kwadrat ta' \sqrt{5} huwa 5.

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

Biex timmultiplika \sqrt{5} u \sqrt{3}, immultiplika n-numri taħt l-għerq kwadrat.

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

Il-kwadrat ta' \sqrt{3} huwa 3.

$$4+\sqrt{15}+\frac{5-2\sqrt{15}+3}{2}$$

Żid 5 u 3 biex tikseb 8.

$$4+\sqrt{15}+\frac{8-2\sqrt{15}}{2}$$

Iddividi kull terminu ta' 8-2\sqrt{15} b'2 biex tikseb4-\sqrt{15}.

$$4+\sqrt{15}+4-\sqrt{15}$$

Żid 4 u 4 biex tikseb 8.

$$8+\sqrt{15}-\sqrt{15}$$

Ikkombina \sqrt{15} u -\sqrt{15} biex tikseb 0.

$$8$$

Fattur

$$2^{3}$$