Rešite 12/9+sqrt{7}

$$\frac { 12 } { 9 + \sqrt { 7 } }$$

Ovrednoti

$$\frac{54-6\sqrt{7}}{37}\approx 1,030418706$$

Koraki rešitve

Racionalizirajte imenovalec \frac{12}{9+\sqrt{7}} tako, da pomnožite števec in imenovalec s 9-\sqrt{7}.

$$\frac{12\left(9-\sqrt{7}\right)}{\left(9+\sqrt{7}\right)\left(9-\sqrt{7}\right)}$$

Razmislite o \left(9+\sqrt{7}\right)\left(9-\sqrt{7}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{12\left(9-\sqrt{7}\right)}{9^{2}-\left(\sqrt{7}\right)^{2}}$$

Kvadrat števila 9. Kvadrat števila \sqrt{7}.

$$\frac{12\left(9-\sqrt{7}\right)}{81-7}$$

Odštejte 7 od 81, da dobite 74.

$$\frac{12\left(9-\sqrt{7}\right)}{74}$$

Delite 12\left(9-\sqrt{7}\right) s/z 74, da dobite \frac{6}{37}\left(9-\sqrt{7}\right).

$$\frac{6}{37}\left(9-\sqrt{7}\right)$$

Uporabite distributivnost, da pomnožite \frac{6}{37} s/z 9-\sqrt{7}.

$$\frac{6}{37}\times 9+\frac{6}{37}\left(-1\right)\sqrt{7}$$

Izrazite \frac{6}{37}\times 9 kot enojni ulomek.

$$\frac{6\times 9}{37}+\frac{6}{37}\left(-1\right)\sqrt{7}$$

Pomnožite 6 in 9, da dobite 54.

$$\frac{54}{37}+\frac{6}{37}\left(-1\right)\sqrt{7}$$

Pomnožite \frac{6}{37} in -1, da dobite -\frac{6}{37}.

$$\frac{54}{37}-\frac{6}{37}\sqrt{7}$$