Rešite (frac{sqrt{2}}{sqrt{2}-18})^2

$$( \frac { \sqrt { 2 } } { \sqrt { 2 } - 18 } ) ^ { 2 }$$

Ovrednoti

$$\frac{18\sqrt{2}+163}{25921}\approx 0,007270393$$

Koraki rešitve

Racionalizirajte imenovalec \frac{\sqrt{2}}{\sqrt{2}-18} tako, da pomnožite števec in imenovalec s \sqrt{2}+18.

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

Razmislite o \left(\sqrt{2}-18\right)\left(\sqrt{2}+18\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Kvadrat števila \sqrt{2}. Kvadrat števila 18.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{2-324}\right)^{2}$$

Odštejte 324 od 2, da dobite -322.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322}\right)^{2}$$

Če želite dobiti potenco vrednosti \frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322}, potencirajte števec in imenovalec, nato pa delite.

$$\frac{\left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}}{\left(-322\right)^{2}}$$

Razčlenite \left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}.

$$\frac{\left(\sqrt{2}\right)^{2}\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

Kvadrat vrednosti \sqrt{2} je 2.

$$\frac{2\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

Uporabite binomski izrek \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, da razširite \left(\sqrt{2}+18\right)^{2}.

$$\frac{2\left(\left(\sqrt{2}\right)^{2}+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

Kvadrat vrednosti \sqrt{2} je 2.

$$\frac{2\left(2+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

Seštejte 2 in 324, da dobite 326.

$$\frac{2\left(326+36\sqrt{2}\right)}{\left(-322\right)^{2}}$$

Izračunajte potenco -322 števila 2, da dobite 103684.

$$\frac{2\left(326+36\sqrt{2}\right)}{103684}$$

Delite 2\left(326+36\sqrt{2}\right) s/z 103684, da dobite \frac{1}{51842}\left(326+36\sqrt{2}\right).

$$\frac{1}{51842}\left(326+36\sqrt{2}\right)$$

Uporabite distributivnost, da pomnožite \frac{1}{51842} s/z 326+36\sqrt{2}.

$$\frac{163}{25921}+\frac{18}{25921}\sqrt{2}$$

Razširi

$$\frac{18 \sqrt{2} + 163}{25921} = 0,007270392505023561$$

Koraki rešitve

Racionalizirajte imenovalec \frac{\sqrt{2}}{\sqrt{2}-18} tako, da pomnožite števec in imenovalec s \sqrt{2}+18.

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

Razmislite o \left(\sqrt{2}-18\right)\left(\sqrt{2}+18\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Kvadrat števila \sqrt{2}. Kvadrat števila 18.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{2-324}\right)^{2}$$

Odštejte 324 od 2, da dobite -322.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322}\right)^{2}$$

Če želite dobiti potenco vrednosti \frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322}, potencirajte števec in imenovalec, nato pa delite.

$$\frac{\left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}}{\left(-322\right)^{2}}$$

Razčlenite \left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}.

$$\frac{\left(\sqrt{2}\right)^{2}\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

Kvadrat vrednosti \sqrt{2} je 2.

$$\frac{2\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

Uporabite binomski izrek \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, da razširite \left(\sqrt{2}+18\right)^{2}.

$$\frac{2\left(\left(\sqrt{2}\right)^{2}+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

Kvadrat vrednosti \sqrt{2} je 2.

$$\frac{2\left(2+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

Seštejte 2 in 324, da dobite 326.

$$\frac{2\left(326+36\sqrt{2}\right)}{\left(-322\right)^{2}}$$

Izračunajte potenco -322 števila 2, da dobite 103684.

$$\frac{2\left(326+36\sqrt{2}\right)}{103684}$$

Delite 2\left(326+36\sqrt{2}\right) s/z 103684, da dobite \frac{1}{51842}\left(326+36\sqrt{2}\right).

$$\frac{1}{51842}\left(326+36\sqrt{2}\right)$$

Uporabite distributivnost, da pomnožite \frac{1}{51842} s/z 326+36\sqrt{2}.

$$\frac{163}{25921}+\frac{18}{25921}\sqrt{2}$$