Issolvi frac{sqrt{7}+sqrt{5}}{sqrt{7}-sqrt{5}}+frac{sqrt{7}-sqrt{5}}{sqrt{7}+sqrt{5}}=

$$\frac { \sqrt { 7 } + \sqrt { 5 } } { \sqrt { 7 } - \sqrt { 5 } } + \frac { \sqrt { 7 } - \sqrt { 5 } } { \sqrt { 7 } + \sqrt { 5 } } =$$

Evalwa

$$12$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{7}+\sqrt{5}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Ikkunsidra li \left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\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}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Ikkwadra \sqrt{7}. Ikkwadra \sqrt{5}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{7-5}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Naqqas 5 minn 7 biex tikseb 2.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Immultiplika \sqrt{7}+\sqrt{5} u \sqrt{7}+\sqrt{5} biex tikseb \left(\sqrt{7}+\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{7}+\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(\sqrt{7}+\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{7}\right)^{2}+2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Il-kwadrat ta' \sqrt{7} huwa 7.

$$\frac{7+2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Biex timmultiplika \sqrt{7} u \sqrt{5}, immultiplika n-numri taħt l-għerq kwadrat.

$$\frac{7+2\sqrt{35}+\left(\sqrt{5}\right)^{2}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

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

$$\frac{7+2\sqrt{35}+5}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Żid 7 u 5 biex tikseb 12.

$$\frac{12+2\sqrt{35}}{2}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Iddividi kull terminu ta' 12+2\sqrt{35} b'2 biex tikseb6+\sqrt{35}.

$$6+\sqrt{35}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$$

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{7}-\sqrt{5}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}$$

Ikkunsidra li \left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\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}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Ikkwadra \sqrt{7}. Ikkwadra \sqrt{5}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{7-5}$$

Naqqas 5 minn 7 biex tikseb 2.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{2}$$

Immultiplika \sqrt{7}-\sqrt{5} u \sqrt{7}-\sqrt{5} biex tikseb \left(\sqrt{7}-\sqrt{5}\right)^{2}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}-\sqrt{5}\right)^{2}}{2}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(\sqrt{7}-\sqrt{5}\right)^{2}.

$$6+\sqrt{35}+\frac{\left(\sqrt{7}\right)^{2}-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}$$

Il-kwadrat ta' \sqrt{7} huwa 7.

$$6+\sqrt{35}+\frac{7-2\sqrt{7}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2}$$

Biex timmultiplika \sqrt{7} u \sqrt{5}, immultiplika n-numri taħt l-għerq kwadrat.

$$6+\sqrt{35}+\frac{7-2\sqrt{35}+\left(\sqrt{5}\right)^{2}}{2}$$

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

$$6+\sqrt{35}+\frac{7-2\sqrt{35}+5}{2}$$

Żid 7 u 5 biex tikseb 12.

$$6+\sqrt{35}+\frac{12-2\sqrt{35}}{2}$$

Iddividi kull terminu ta' 12-2\sqrt{35} b'2 biex tikseb6-\sqrt{35}.

$$6+\sqrt{35}+6-\sqrt{35}$$

Żid 6 u 6 biex tikseb 12.

$$12+\sqrt{35}-\sqrt{35}$$

Ikkombina \sqrt{35} u -\sqrt{35} biex tikseb 0.

$$12$$

Fattur

$$2^{2}\times 3$$