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

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

Evalwa

$$\sqrt{35}+6\approx 11.916079783$$

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}.

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

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}.

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

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

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

Naqqas 5 minn 7 biex tikseb 2.

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

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

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

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

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

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

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

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

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

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

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

Żid 7 u 5 biex tikseb 12.

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

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

$$6+\sqrt{35}$$