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

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

Evaluate

$$\frac{3\sqrt{5}}{4}-\frac{\sqrt{3}}{2}-\frac{\sqrt{15}}{2}+\frac{3}{4}\approx -0.375466094$$

Solution Steps

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

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

Consider \left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

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

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

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

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

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

Factor

$$\frac{3 \sqrt{5} + 3 - 2 \sqrt{3} - 2 \sqrt{15}}{4} = -0.37546609376330475$$