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

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

Ovrednoti

$$\frac{1}{2}=0,5$$

Koraki rešitve

Racionalizirajte imenovalec \frac{3+2\sqrt{2}}{2+\sqrt{2}} tako, da pomnožite števec in imenovalec s 2-\sqrt{2}.

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

Razmislite o \left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Kvadrat števila 2. Kvadrat števila \sqrt{2}.

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

Odštejte 2 od 4, da dobite 2.

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

Racionalizirajte imenovalec \frac{\sqrt{2}-1}{\sqrt{2}} tako, da pomnožite števec in imenovalec s \sqrt{2}.

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

Kvadrat vrednosti \sqrt{2} je 2.

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

Pomnožite \frac{\left(3+2\sqrt{2}\right)\left(2-\sqrt{2}\right)}{2} s/z \frac{\left(\sqrt{2}-1\right)\sqrt{2}}{2} tako, da pomnožite števec s števcem in imenovalec z imenovalcem.

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

Pomnožite 2 in 2, da dobite 4.

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

Uporabite distributivnost tako, da pomnožite vsako vrednost 3+2\sqrt{2} z vsako vrednostjo 2-\sqrt{2}.

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

Združite -3\sqrt{2} in 4\sqrt{2}, da dobite \sqrt{2}.

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

Kvadrat vrednosti \sqrt{2} je 2.

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

Pomnožite -2 in 2, da dobite -4.

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

Odštejte 4 od 6, da dobite 2.

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

Uporabite distributivnost tako, da pomnožite vsako vrednost 2+\sqrt{2} z vsako vrednostjo \sqrt{2}-1.

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

Kvadrat vrednosti \sqrt{2} je 2.

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

Seštejte -2 in 2, da dobite 0.

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

Združite 2\sqrt{2} in -\sqrt{2}, da dobite \sqrt{2}.

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

Pomnožite \sqrt{2} in \sqrt{2}, da dobite 2.

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

Zmanjšajte ulomek \frac{2}{4} na najmanjši imenovalec tako, da izpeljete in okrajšate 2.

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

Faktoriziraj

$$\frac{1}{2} = 0,5$$