Issolvi frac{21sqrt{15}}{512+5sqrt{3}}

$$\frac { 21 \sqrt { 15 } } { 512 + 5 \sqrt { 3 } }$$

Evalwa

$$\frac{10752\sqrt{15}-315\sqrt{5}}{262069}\approx 0.156210599$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{21\sqrt{15}}{512+5\sqrt{3}} billi timmultiplika in-numeratur u d-denominatur mill-512-5\sqrt{3}.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{\left(512+5\sqrt{3}\right)\left(512-5\sqrt{3}\right)}$$

Ikkunsidra li \left(512+5\sqrt{3}\right)\left(512-5\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}.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{512^{2}-\left(5\sqrt{3}\right)^{2}}$$

Ikkalkula 512 bil-power ta' 2 u tikseb 262144.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-\left(5\sqrt{3}\right)^{2}}$$

Espandi \left(5\sqrt{3}\right)^{2}.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-5^{2}\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-25\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-25\times 3}$$

Immultiplika 25 u 3 biex tikseb 75.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-75}$$

Naqqas 75 minn 262144 biex tikseb 262069.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262069}$$

Uża l-propjetà distributtiva biex timmultiplika 21\sqrt{15} b'512-5\sqrt{3}.

$$\frac{10752\sqrt{15}-105\sqrt{3}\sqrt{15}}{262069}$$

Iffattura 15=3\times 5. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{3\times 5} bħala l-prodott tal-għeruq kwadrati \sqrt{3}\sqrt{5}.

$$\frac{10752\sqrt{15}-105\sqrt{3}\sqrt{3}\sqrt{5}}{262069}$$

Immultiplika \sqrt{3} u \sqrt{3} biex tikseb 3.

$$\frac{10752\sqrt{15}-105\times 3\sqrt{5}}{262069}$$

Immultiplika -105 u 3 biex tikseb -315.

$$\frac{10752\sqrt{15}-315\sqrt{5}}{262069}$$