Solve frac{3-sqrt{5}}{4+sqrt{5}}

$$\frac { 3 - \sqrt { 5 } } { 4 + \sqrt { 5 } }$$

Evaluate

$$\frac{17-7\sqrt{5}}{11}\approx 0.122502196$$

Solution Steps

Rationalize the denominator of \frac{3-\sqrt{5}}{4+\sqrt{5}} by multiplying numerator and denominator by 4-\sqrt{5}.

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

Consider \left(4+\sqrt{5}\right)\left(4-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 4. Square \sqrt{5}.

$$\frac{\left(3-\sqrt{5}\right)\left(4-\sqrt{5}\right)}{16-5}$$

Subtract 5 from 16 to get 11.

$$\frac{\left(3-\sqrt{5}\right)\left(4-\sqrt{5}\right)}{11}$$

Apply the distributive property by multiplying each term of 3-\sqrt{5} by each term of 4-\sqrt{5}.

$$\frac{12-3\sqrt{5}-4\sqrt{5}+\left(\sqrt{5}\right)^{2}}{11}$$

Combine -3\sqrt{5} and -4\sqrt{5} to get -7\sqrt{5}.

$$\frac{12-7\sqrt{5}+\left(\sqrt{5}\right)^{2}}{11}$$

The square of \sqrt{5} is 5.

$$\frac{12-7\sqrt{5}+5}{11}$$

Add 12 and 5 to get 17.

$$\frac{17-7\sqrt{5}}{11}$$