Solve frac{2sqrt{5}-4sqrt{3}}{-1+sqrt{3}}

$$\frac{ 2 \sqrt{ 5 } -4 \sqrt{ 3 } }{ -1+ \sqrt{ 3 } }$$

Evaluate

$$\sqrt{5}+\sqrt{15}-2\sqrt{3}-6\approx -3.355050291$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{5}-4\sqrt{3}}{-1+\sqrt{3}} by multiplying numerator and denominator by -1-\sqrt{3}.

$$\frac{\left(2\sqrt{5}-4\sqrt{3}\right)\left(-1-\sqrt{3}\right)}{\left(-1+\sqrt{3}\right)\left(-1-\sqrt{3}\right)}$$

Consider \left(-1+\sqrt{3}\right)\left(-1-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{5}-4\sqrt{3}\right)\left(-1-\sqrt{3}\right)}{\left(-1\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square -1. Square \sqrt{3}.

$$\frac{\left(2\sqrt{5}-4\sqrt{3}\right)\left(-1-\sqrt{3}\right)}{1-3}$$

Subtract 3 from 1 to get -2.

$$\frac{\left(2\sqrt{5}-4\sqrt{3}\right)\left(-1-\sqrt{3}\right)}{-2}$$

Apply the distributive property by multiplying each term of 2\sqrt{5}-4\sqrt{3} by each term of -1-\sqrt{3}.

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

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

$$\frac{-2\sqrt{5}-2\sqrt{15}+4\sqrt{3}+4\left(\sqrt{3}\right)^{2}}{-2}$$

The square of \sqrt{3} is 3.

$$\frac{-2\sqrt{5}-2\sqrt{15}+4\sqrt{3}+4\times 3}{-2}$$

Multiply 4 and 3 to get 12.

$$\frac{-2\sqrt{5}-2\sqrt{15}+4\sqrt{3}+12}{-2}$$

Divide each term of -2\sqrt{5}-2\sqrt{15}+4\sqrt{3}+12 by -2 to get \sqrt{5}+\sqrt{15}-2\sqrt{3}-6.

$$\sqrt{5}+\sqrt{15}-2\sqrt{3}-6$$

Factor

$$\sqrt{5} + \sqrt{15} - 2 \sqrt{3} - 6 = -3.355050291$$