Solve frac{2+sqrt{3}}{4-sqrt{5}}*4

$$\frac { 2 + \sqrt { 3 } } { 4 - \sqrt { 5 } } \cdot 4$$

Evaluate

$$\frac{4\left(\sqrt{15}+2\sqrt{5}+4\sqrt{3}+8\right)}{11}\approx 8.463026375$$

Solution Steps

Rationalize the denominator of \frac{2+\sqrt{3}}{4-\sqrt{5}} by multiplying numerator and denominator by 4+\sqrt{5}.

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

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}.

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

Square 4. Square \sqrt{5}.

$$4\times \frac{\left(2+\sqrt{3}\right)\left(4+\sqrt{5}\right)}{16-5}$$

Subtract 5 from 16 to get 11.

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

Express 4\times \frac{\left(2+\sqrt{3}\right)\left(4+\sqrt{5}\right)}{11} as a single fraction.

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

Use the distributive property to multiply 4 by 2+\sqrt{3}.

$$\frac{\left(8+4\sqrt{3}\right)\left(4+\sqrt{5}\right)}{11}$$

Apply the distributive property by multiplying each term of 8+4\sqrt{3} by each term of 4+\sqrt{5}.

$$\frac{32+8\sqrt{5}+16\sqrt{3}+4\sqrt{3}\sqrt{5}}{11}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{32+8\sqrt{5}+16\sqrt{3}+4\sqrt{15}}{11}$$