Issolvi 2/sqrt{2-2}+frac{sqrt{2}+1}{sqrt{2}-1}-frac{sqrt{32}}{2}

$$\frac { 2 } { \sqrt { 2 } - 2 } + \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 } - \frac { \sqrt { 32 } } { 2 }$$

Evalwa

$$1-\sqrt{2}\approx -0.414213562$$

Passi tas-Soluzzjonijiet

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

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}-2\right)\left(\sqrt{2}+2\right)}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

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

$$\frac{2\left(\sqrt{2}+2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Ikkwadra \sqrt{2}. Ikkwadra 2.

$$\frac{2\left(\sqrt{2}+2\right)}{2-4}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Naqqas 4 minn 2 biex tikseb -2.

$$\frac{2\left(\sqrt{2}+2\right)}{-2}+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Annulla -2 u -2.

$$-\left(\sqrt{2}+2\right)+\frac{\sqrt{2}+1}{\sqrt{2}-1}-\frac{\sqrt{32}}{2}$$

Irrazzjonalizza d-denominatur tal-\frac{\sqrt{2}+1}{\sqrt{2}-1} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{2}+1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}-\frac{\sqrt{32}}{2}$$

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}.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}-\frac{\sqrt{32}}{2}$$

Ikkwadra \sqrt{2}. Ikkwadra 1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{2-1}-\frac{\sqrt{32}}{2}$$

Naqqas 1 minn 2 biex tikseb 1.

$$-\left(\sqrt{2}+2\right)+\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)}{1}-\frac{\sqrt{32}}{2}$$

Kwalunkwe ħaġa diviża b'wieħed tagħti riżultat tagħha stess.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)\left(\sqrt{2}+1\right)-\frac{\sqrt{32}}{2}$$

Immultiplika \sqrt{2}+1 u \sqrt{2}+1 biex tikseb \left(\sqrt{2}+1\right)^{2}.

$$-\left(\sqrt{2}+2\right)+\left(\sqrt{2}+1\right)^{2}-\frac{\sqrt{32}}{2}$$

Iffattura 32=4^{2}\times 2. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{4^{2}\times 2} bħala l-prodott tal-għeruq kwadrati \sqrt{4^{2}}\sqrt{2}. Ħu l-għerq kwadrat ta' 4^{2}.

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

Iddividi 4\sqrt{2} b'2 biex tikseb2\sqrt{2}.

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

Biex issib l-oppost ta' \sqrt{2}+2, sib l-oppost ta' kull terminu.

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

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

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

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

$$-\sqrt{2}-2+2+2\sqrt{2}+1-2\sqrt{2}$$

Żid 2 u 1 biex tikseb 3.

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

Żid -2 u 3 biex tikseb 1.

$$-\sqrt{2}+1+2\sqrt{2}-2\sqrt{2}$$

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

$$\sqrt{2}+1-2\sqrt{2}$$

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

$$-\sqrt{2}+1$$

Fattur

$$1-\sqrt{2}$$