Issolvi (sqrt{3}+1)^2-(sqrt{2}+1)(sqrt{2}-1)+2(3-sqrt{3})

$$( \sqrt { 3 } + 1 ) ^ { 2 } - ( \sqrt { 2 } + 1 ) ( \sqrt { 2 } - 1 ) + 2 ( 3 - \sqrt { 3 } )$$

Evalwa

$$9$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(\sqrt{3}+1\right)^{2}.

$$\left(\sqrt{3}\right)^{2}+2\sqrt{3}+1-\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)+2\left(3-\sqrt{3}\right)$$

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

$$3+2\sqrt{3}+1-\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)+2\left(3-\sqrt{3}\right)$$

Żid 3 u 1 biex tikseb 4.

$$4+2\sqrt{3}-\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)+2\left(3-\sqrt{3}\right)$$

Ikkunsidra li \left(\sqrt{2}+1\right)\left(\sqrt{2}-1\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}. Ikkwadra 1.

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

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

$$4+2\sqrt{3}-\left(2-1\right)+2\left(3-\sqrt{3}\right)$$

Naqqas 1 minn 2 biex tikseb 1.

$$4+2\sqrt{3}-1+2\left(3-\sqrt{3}\right)$$

Naqqas 1 minn 4 biex tikseb 3.

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

Uża l-propjetà distributtiva biex timmultiplika 2 b'3-\sqrt{3}.

$$3+2\sqrt{3}+6-2\sqrt{3}$$

Żid 3 u 6 biex tikseb 9.

$$9+2\sqrt{3}-2\sqrt{3}$$

Ikkombina 2\sqrt{3} u -2\sqrt{3} biex tikseb 0.

$$9$$

Fattur

$$3^{2}$$