Issolvi frac{2sqrt{3}-5}{2sqrt{3}+5}

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

Evalwa

$$\frac{20\sqrt{3}-37}{13}\approx -0.181460296$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{2\sqrt{3}-5}{2\sqrt{3}+5} billi timmultiplika in-numeratur u d-denominatur mill-2\sqrt{3}-5.

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

Ikkunsidra li \left(2\sqrt{3}+5\right)\left(2\sqrt{3}-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}.

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

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

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

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

$$\frac{4\left(\sqrt{3}\right)^{2}-20\sqrt{3}+25}{\left(2\sqrt{3}\right)^{2}-5^{2}}$$

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

$$\frac{4\times 3-20\sqrt{3}+25}{\left(2\sqrt{3}\right)^{2}-5^{2}}$$

Immultiplika 4 u 3 biex tikseb 12.

$$\frac{12-20\sqrt{3}+25}{\left(2\sqrt{3}\right)^{2}-5^{2}}$$

Żid 12 u 25 biex tikseb 37.

$$\frac{37-20\sqrt{3}}{\left(2\sqrt{3}\right)^{2}-5^{2}}$$

Espandi \left(2\sqrt{3}\right)^{2}.

$$\frac{37-20\sqrt{3}}{2^{2}\left(\sqrt{3}\right)^{2}-5^{2}}$$

Ikkalkula 2 bil-power ta' 2 u tikseb 4.

$$\frac{37-20\sqrt{3}}{4\left(\sqrt{3}\right)^{2}-5^{2}}$$

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

$$\frac{37-20\sqrt{3}}{4\times 3-5^{2}}$$

Immultiplika 4 u 3 biex tikseb 12.

$$\frac{37-20\sqrt{3}}{12-5^{2}}$$

Ikkalkula 5 bil-power ta' 2 u tikseb 25.

$$\frac{37-20\sqrt{3}}{12-25}$$

Naqqas 25 minn 12 biex tikseb -13.

$$\frac{37-20\sqrt{3}}{-13}$$

Immultiplika kemm in-numeratur u kif ukoll id-denominatur b’-1.

$$\frac{-37+20\sqrt{3}}{13}$$