Solve frac{6sqrt{5}}{3+sqrt{5}+sqrt{4}}

$$\frac { 6 \sqrt { 5 } } { 3 + \sqrt { 5 } + \sqrt { 4 } }$$

Evaluate

$$\frac{3\left(\sqrt{5}-1\right)}{2}\approx 1.854101966$$

Solution Steps

Calculate the square root of 4 and get 2.

$$\frac{6\sqrt{5}}{3+\sqrt{5}+2}$$

Add 3 and 2 to get 5.

$$\frac{6\sqrt{5}}{5+\sqrt{5}}$$

Rationalize the denominator of \frac{6\sqrt{5}}{5+\sqrt{5}} by multiplying numerator and denominator by 5-\sqrt{5}.

$$\frac{6\sqrt{5}\left(5-\sqrt{5}\right)}{\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)}$$

Consider \left(5+\sqrt{5}\right)\left(5-\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}.

$$\frac{6\sqrt{5}\left(5-\sqrt{5}\right)}{5^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 5. Square \sqrt{5}.

$$\frac{6\sqrt{5}\left(5-\sqrt{5}\right)}{25-5}$$

Subtract 5 from 25 to get 20.

$$\frac{6\sqrt{5}\left(5-\sqrt{5}\right)}{20}$$

Use the distributive property to multiply 6\sqrt{5} by 5-\sqrt{5}.

$$\frac{30\sqrt{5}-6\left(\sqrt{5}\right)^{2}}{20}$$

The square of \sqrt{5} is 5.

$$\frac{30\sqrt{5}-6\times 5}{20}$$

Multiply -6 and 5 to get -30.

$$\frac{30\sqrt{5}-30}{20}$$