Rešite frac{7+3sqrt{5}}{3+sqrt{5}}-frac{7-3sqrt{5}}{3-sqrt{5}}

$$\frac { 7 + 3 \sqrt { 5 } } { 3 + \sqrt { 5 } } - \frac { 7 - 3 \sqrt { 5 } } { 3 - \sqrt { 5 } }$$

Ovrednoti

$$\sqrt{5}\approx 2,236067977$$

Kratka navodila za rešitev

Racionalizirajte imenovalec \frac{7+3\sqrt{5}}{3+\sqrt{5}} tako, da pomnožite števec in imenovalec s 3-\sqrt{5}.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}-\frac{7-3\sqrt{5}}{3-\sqrt{5}}$$

Razmislite o \left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}-\frac{7-3\sqrt{5}}{3-\sqrt{5}}$$

Kvadrat števila 3. Kvadrat števila \sqrt{5}.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{9-5}-\frac{7-3\sqrt{5}}{3-\sqrt{5}}$$

Odštejte 5 od 9, da dobite 4.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{4}-\frac{7-3\sqrt{5}}{3-\sqrt{5}}$$

Racionalizirajte imenovalec \frac{7-3\sqrt{5}}{3-\sqrt{5}} tako, da pomnožite števec in imenovalec s 3+\sqrt{5}.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{4}-\frac{\left(7-3\sqrt{5}\right)\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}$$

Razmislite o \left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{4}-\frac{\left(7-3\sqrt{5}\right)\left(3+\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}$$

Kvadrat števila 3. Kvadrat števila \sqrt{5}.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{4}-\frac{\left(7-3\sqrt{5}\right)\left(3+\sqrt{5}\right)}{9-5}$$

Odštejte 5 od 9, da dobite 4.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{4}-\frac{\left(7-3\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4}$$

Ker \frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)}{4} in \frac{\left(7-3\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

$$\frac{\left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)-\left(7-3\sqrt{5}\right)\left(3+\sqrt{5}\right)}{4}$$

Izvedi množenje v \left(7+3\sqrt{5}\right)\left(3-\sqrt{5}\right)-\left(7-3\sqrt{5}\right)\left(3+\sqrt{5}\right).

$$\frac{21-7\sqrt{5}+9\sqrt{5}-15-21-7\sqrt{5}+9\sqrt{5}+15}{4}$$

Izvedi izračune v 21-7\sqrt{5}+9\sqrt{5}-15-21-7\sqrt{5}+9\sqrt{5}+15.

$$\frac{4\sqrt{5}}{4}$$

Okrajšaj 4 in 4.

$$\sqrt{5}$$