Solve frac{8+4-2sqrt{5}-4sqrt{5}+2sqrt{10}}{1-sqrt{5}}

$$\frac { 8 + 4 - 2 \sqrt { 5 } - 4 \sqrt { 5 } + 2 \sqrt { 10 } } { 1 - \sqrt { 5 } }$$

Evaluate

$$\frac{9-\sqrt{10}-3\sqrt{5}-5\sqrt{2}}{2}\approx -3.970774702$$

Solution Steps

Add 8 and 4 to get 12.

$$\frac{12-2\sqrt{5}-4\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

Combine -2\sqrt{5} and -4\sqrt{5} to get -6\sqrt{5}.

$$\frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}}$$

Rationalize the denominator of \frac{12-6\sqrt{5}+2\sqrt{10}}{1-\sqrt{5}} by multiplying numerator and denominator by 1+\sqrt{5}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}$$

Consider \left(1-\sqrt{5}\right)\left(1+\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}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 1. Square \sqrt{5}.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{1-5}$$

Subtract 5 from 1 to get -4.

$$\frac{\left(12-6\sqrt{5}+2\sqrt{10}\right)\left(1+\sqrt{5}\right)}{-4}$$

Apply the distributive property by multiplying each term of 12-6\sqrt{5}+2\sqrt{10} by each term of 1+\sqrt{5}.

$$\frac{12+12\sqrt{5}-6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Combine 12\sqrt{5} and -6\sqrt{5} to get 6\sqrt{5}.

$$\frac{12+6\sqrt{5}-6\left(\sqrt{5}\right)^{2}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

The square of \sqrt{5} is 5.

$$\frac{12+6\sqrt{5}-6\times 5+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Multiply -6 and 5 to get -30.

$$\frac{12+6\sqrt{5}-30+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Subtract 30 from 12 to get -18.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{10}\sqrt{5}}{-4}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\sqrt{5}\sqrt{2}\sqrt{5}}{-4}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+2\times 5\sqrt{2}}{-4}$$

Multiply 2 and 5 to get 10.

$$\frac{-18+6\sqrt{5}+2\sqrt{10}+10\sqrt{2}}{-4}$$

Factor

$$\frac{9 - \sqrt{10} - 3 \sqrt{5} - 5 \sqrt{2}}{2} = -3.9707747022666124$$