Issolvi frac{4-sqrt{2}}{4+sqrt{2}}-frac{4+sqrt{2}}{4-sqrt{2}}

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

Evalwa

$$-\frac{8\sqrt{2}}{7}\approx -1.616244071$$

Passi Qosra tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{4-\sqrt{2}}{4+\sqrt{2}} billi timmultiplika in-numeratur u d-denominatur mill-4-\sqrt{2}.

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

Ikkunsidra li \left(4+\sqrt{2}\right)\left(4-\sqrt{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}.

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

Ikkwadra 4. Ikkwadra \sqrt{2}.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{16-2}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Naqqas 2 minn 16 biex tikseb 14.

$$\frac{\left(4-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Immultiplika 4-\sqrt{2} u 4-\sqrt{2} biex tikseb \left(4-\sqrt{2}\right)^{2}.

$$\frac{\left(4-\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Uża teorema binomjali \left(a-b\right)^{2}=a^{2}-2ab+b^{2} biex tespandi \left(4-\sqrt{2}\right)^{2}.

$$\frac{16-8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

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

$$\frac{16-8\sqrt{2}+2}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Żid 16 u 2 biex tikseb 18.

$$\frac{18-8\sqrt{2}}{14}-\frac{4+\sqrt{2}}{4-\sqrt{2}}$$

Irrazzjonalizza d-denominatur tal-\frac{4+\sqrt{2}}{4-\sqrt{2}} billi timmultiplika in-numeratur u d-denominatur mill-4+\sqrt{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{\left(4-\sqrt{2}\right)\left(4+\sqrt{2}\right)}$$

Ikkunsidra li \left(4-\sqrt{2}\right)\left(4+\sqrt{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}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{4^{2}-\left(\sqrt{2}\right)^{2}}$$

Ikkwadra 4. Ikkwadra \sqrt{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{16-2}$$

Naqqas 2 minn 16 biex tikseb 14.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)\left(4+\sqrt{2}\right)}{14}$$

Immultiplika 4+\sqrt{2} u 4+\sqrt{2} biex tikseb \left(4+\sqrt{2}\right)^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{\left(4+\sqrt{2}\right)^{2}}{14}$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(4+\sqrt{2}\right)^{2}.

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}$$

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

$$\frac{18-8\sqrt{2}}{14}-\frac{16+8\sqrt{2}+2}{14}$$

Żid 16 u 2 biex tikseb 18.

$$\frac{18-8\sqrt{2}}{14}-\frac{18+8\sqrt{2}}{14}$$

Billi \frac{18-8\sqrt{2}}{14} u \frac{18+8\sqrt{2}}{14} għandhom l-istess denominatur, naqqashom billi tnaqqas in-numeraturi tagħhom.

$$\frac{18-8\sqrt{2}-\left(18+8\sqrt{2}\right)}{14}$$

Agħmel il-multiplikazzjonijiet fi 18-8\sqrt{2}-\left(18+8\sqrt{2}\right).

$$\frac{18-8\sqrt{2}-18-8\sqrt{2}}{14}$$

Agħmel il-kalkoli fi 18-8\sqrt{2}-18-8\sqrt{2}.

$$\frac{-16\sqrt{2}}{14}$$

Iddividi -16\sqrt{2} b'14 biex tikseb-\frac{8}{7}\sqrt{2}.

$$-\frac{8}{7}\sqrt{2}$$