Issolvi (sqrt{2}-2)^2+sqrt{12/3}divsqrt{5/24}

$$( \sqrt { 2 } - 2 ) ^ { 2 } + \sqrt { 1 \frac { 2 } { 3 } } \div \sqrt { \frac { 5 } { 24 } }$$

Evalwa

$$6-2\sqrt{2}\approx 3.171572875$$

Passi tas-Soluzzjonijiet

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

$$\left(\sqrt{2}\right)^{2}-4\sqrt{2}+4+\frac{\sqrt{\frac{1\times 3+2}{3}}}{\sqrt{\frac{5}{24}}}$$

Il-kwadrat ta' \sqrt{2} huwa 2.

$$2-4\sqrt{2}+4+\frac{\sqrt{\frac{1\times 3+2}{3}}}{\sqrt{\frac{5}{24}}}$$

Żid 2 u 4 biex tikseb 6.

$$6-4\sqrt{2}+\frac{\sqrt{\frac{1\times 3+2}{3}}}{\sqrt{\frac{5}{24}}}$$

Immultiplika 1 u 3 biex tikseb 3.

$$6-4\sqrt{2}+\frac{\sqrt{\frac{3+2}{3}}}{\sqrt{\frac{5}{24}}}$$

Żid 3 u 2 biex tikseb 5.

$$6-4\sqrt{2}+\frac{\sqrt{\frac{5}{3}}}{\sqrt{\frac{5}{24}}}$$

Erġa' ikteb id-diviżjoni tal-għerq kwadrat \sqrt{\frac{5}{3}} bħala d-diviżjoni tal-għeruq kwadrati \frac{\sqrt{5}}{\sqrt{3}}.

$$6-4\sqrt{2}+\frac{\frac{\sqrt{5}}{\sqrt{3}}}{\sqrt{\frac{5}{24}}}$$

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

$$6-4\sqrt{2}+\frac{\frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{\frac{5}{24}}}$$

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

$$6-4\sqrt{2}+\frac{\frac{\sqrt{5}\sqrt{3}}{3}}{\sqrt{\frac{5}{24}}}$$

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

$$6-4\sqrt{2}+\frac{\frac{\sqrt{15}}{3}}{\sqrt{\frac{5}{24}}}$$

Erġa' ikteb id-diviżjoni tal-għerq kwadrat \sqrt{\frac{5}{24}} bħala d-diviżjoni tal-għeruq kwadrati \frac{\sqrt{5}}{\sqrt{24}}.

$$6-4\sqrt{2}+\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{5}}{\sqrt{24}}}$$

Iffattura 24=2^{2}\times 6. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{2^{2}\times 6} bħala l-prodott tal-għeruq kwadrati \sqrt{2^{2}}\sqrt{6}. Ħu l-għerq kwadrat ta' 2^{2}.

$$6-4\sqrt{2}+\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{5}}{2\sqrt{6}}}$$

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{5}}{2\sqrt{6}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{6}.

$$6-4\sqrt{2}+\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{5}\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}}$$

Il-kwadrat ta' \sqrt{6} huwa 6.

$$6-4\sqrt{2}+\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{5}\sqrt{6}}{2\times 6}}$$

Biex timmultiplika \sqrt{5} u \sqrt{6}, immultiplika n-numri taħt l-għerq kwadrat.

$$6-4\sqrt{2}+\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{30}}{2\times 6}}$$

Immultiplika 2 u 6 biex tikseb 12.

$$6-4\sqrt{2}+\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{30}}{12}}$$

Iddividi \frac{\sqrt{15}}{3} b'\frac{\sqrt{30}}{12} billi timmultiplika \frac{\sqrt{15}}{3} bir-reċiproku ta' \frac{\sqrt{30}}{12}.

$$6-4\sqrt{2}+\frac{\sqrt{15}\times 12}{3\sqrt{30}}$$

Annulla 3 fin-numeratur u d-denominatur.

$$6-4\sqrt{2}+\frac{4\sqrt{15}}{\sqrt{30}}$$

Irrazzjonalizza d-denominatur tal-\frac{4\sqrt{15}}{\sqrt{30}} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{30}.

$$6-4\sqrt{2}+\frac{4\sqrt{15}\sqrt{30}}{\left(\sqrt{30}\right)^{2}}$$

Il-kwadrat ta' \sqrt{30} huwa 30.

$$6-4\sqrt{2}+\frac{4\sqrt{15}\sqrt{30}}{30}$$

Iffattura 30=15\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{15\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{15}\sqrt{2}.

$$6-4\sqrt{2}+\frac{4\sqrt{15}\sqrt{15}\sqrt{2}}{30}$$

Immultiplika \sqrt{15} u \sqrt{15} biex tikseb 15.

$$6-4\sqrt{2}+\frac{4\times 15\sqrt{2}}{30}$$

Immultiplika 4 u 15 biex tikseb 60.

$$6-4\sqrt{2}+\frac{60\sqrt{2}}{30}$$

Iddividi 60\sqrt{2} b'30 biex tikseb2\sqrt{2}.

$$6-4\sqrt{2}+2\sqrt{2}$$

Ikkombina -4\sqrt{2} u 2\sqrt{2} biex tikseb -2\sqrt{2}.

$$6-2\sqrt{2}$$