Issolvi frac{3+2sqrt{2}}{2+sqrt{2}}*frac{sqrt{2}-1}{sqrt{2}}

$$\frac { 3 + 2 \sqrt { 2 } } { 2 + \sqrt { 2 } } \cdot \frac { \sqrt { 2 } - 1 } { \sqrt { 2 } }$$

Evalwa

$$\frac{1}{2}=0.5$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{3+2\sqrt{2}}{2+\sqrt{2}} billi timmultiplika in-numeratur u d-denominatur mill-2-\sqrt{2}.

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

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

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

Ikkwadra 2. Ikkwadra \sqrt{2}.

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

Naqqas 2 minn 4 biex tikseb 2.

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

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

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

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

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

Immultiplika \frac{\left(3+2\sqrt{2}\right)\left(2-\sqrt{2}\right)}{2} b'\frac{\left(\sqrt{2}-1\right)\sqrt{2}}{2} billi timmultiplika n-numeratur bin-numeratur u d-denominatur bid-denominatur.

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

Immultiplika 2 u 2 biex tikseb 4.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 3+2\sqrt{2} b'kull terminu ta' 2-\sqrt{2}.

$$\frac{\left(6-3\sqrt{2}+4\sqrt{2}-2\left(\sqrt{2}\right)^{2}\right)\left(\sqrt{2}-1\right)\sqrt{2}}{4}$$

Ikkombina -3\sqrt{2} u 4\sqrt{2} biex tikseb \sqrt{2}.

$$\frac{\left(6+\sqrt{2}-2\left(\sqrt{2}\right)^{2}\right)\left(\sqrt{2}-1\right)\sqrt{2}}{4}$$

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

$$\frac{\left(6+\sqrt{2}-2\times 2\right)\left(\sqrt{2}-1\right)\sqrt{2}}{4}$$

Immultiplika -2 u 2 biex tikseb -4.

$$\frac{\left(6+\sqrt{2}-4\right)\left(\sqrt{2}-1\right)\sqrt{2}}{4}$$

Naqqas 4 minn 6 biex tikseb 2.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 2+\sqrt{2} b'kull terminu ta' \sqrt{2}-1.

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

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

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

Żid -2 u 2 biex tikseb 0.

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

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

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

Immultiplika \sqrt{2} u \sqrt{2} biex tikseb 2.

$$\frac{2}{4}$$

Naqqas il-frazzjoni \frac{2}{4} għat-termini l-aktar baxxi billi testratta u tikkanċella barra 2.

$$\frac{1}{2}$$

Fattur

$$\frac{1}{2} = 0.5$$