Issolvi (sqrt{27}-3sqrt{2})^2+(sqrt{3}+2sqrt{2})^2

$$( \sqrt { 27 } - 3 \sqrt { 2 } ) ^ { 2 } + ( \sqrt { 3 } + 2 \sqrt { 2 } ) ^ { 2 }$$

Evalwa

$$56-14\sqrt{6}\approx 21.707143601$$

Passi tas-Soluzzjonijiet

Iffattura 27=3^{2}\times 3. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{3^{2}\times 3} bħala l-prodott tal-għeruq kwadrati \sqrt{3^{2}}\sqrt{3}. Ħu l-għerq kwadrat ta' 3^{2}.

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

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

$$9\left(\sqrt{3}\right)^{2}-18\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

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

$$9\times 3-18\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Immultiplika 9 u 3 biex tikseb 27.

$$27-18\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Biex timmultiplika \sqrt{3} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

$$27-18\sqrt{6}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

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

$$27-18\sqrt{6}+9\times 2+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Immultiplika 9 u 2 biex tikseb 18.

$$27-18\sqrt{6}+18+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Żid 27 u 18 biex tikseb 45.

$$45-18\sqrt{6}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

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

$$45-18\sqrt{6}+\left(\sqrt{3}\right)^{2}+4\sqrt{3}\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

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

$$45-18\sqrt{6}+3+4\sqrt{3}\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Biex timmultiplika \sqrt{3} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

$$45-18\sqrt{6}+3+4\sqrt{6}+4\left(\sqrt{2}\right)^{2}$$

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

$$45-18\sqrt{6}+3+4\sqrt{6}+4\times 2$$

Immultiplika 4 u 2 biex tikseb 8.

$$45-18\sqrt{6}+3+4\sqrt{6}+8$$

Żid 3 u 8 biex tikseb 11.

$$45-18\sqrt{6}+11+4\sqrt{6}$$

Żid 45 u 11 biex tikseb 56.

$$56-18\sqrt{6}+4\sqrt{6}$$

Ikkombina -18\sqrt{6} u 4\sqrt{6} biex tikseb -14\sqrt{6}.

$$56-14\sqrt{6}$$

Espandi

$$56-14\sqrt{6}$$

Passi tas-Soluzzjonijiet

Iffattura 27=3^{2}\times 3. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{3^{2}\times 3} bħala l-prodott tal-għeruq kwadrati \sqrt{3^{2}}\sqrt{3}. Ħu l-għerq kwadrat ta' 3^{2}.

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

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

$$9\left(\sqrt{3}\right)^{2}-18\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

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

$$9\times 3-18\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Immultiplika 9 u 3 biex tikseb 27.

$$27-18\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Biex timmultiplika \sqrt{3} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

$$27-18\sqrt{6}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

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

$$27-18\sqrt{6}+9\times 2+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Immultiplika 9 u 2 biex tikseb 18.

$$27-18\sqrt{6}+18+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

Żid 27 u 18 biex tikseb 45.

$$45-18\sqrt{6}+\left(\sqrt{3}+2\sqrt{2}\right)^{2}$$

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

$$45-18\sqrt{6}+\left(\sqrt{3}\right)^{2}+4\sqrt{3}\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

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

$$45-18\sqrt{6}+3+4\sqrt{3}\sqrt{2}+4\left(\sqrt{2}\right)^{2}$$

Biex timmultiplika \sqrt{3} u \sqrt{2}, immultiplika n-numri taħt l-għerq kwadrat.

$$45-18\sqrt{6}+3+4\sqrt{6}+4\left(\sqrt{2}\right)^{2}$$

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

$$45-18\sqrt{6}+3+4\sqrt{6}+4\times 2$$

Immultiplika 4 u 2 biex tikseb 8.

$$45-18\sqrt{6}+3+4\sqrt{6}+8$$

Żid 3 u 8 biex tikseb 11.

$$45-18\sqrt{6}+11+4\sqrt{6}$$

Żid 45 u 11 biex tikseb 56.

$$56-18\sqrt{6}+4\sqrt{6}$$

Ikkombina -18\sqrt{6} u 4\sqrt{6} biex tikseb -14\sqrt{6}.

$$56-14\sqrt{6}$$