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

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

Evaluate

$$4\sqrt{3}+7\approx 13.92820323$$

Solution Steps

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

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

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}.

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

Square 1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

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

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

Combine 5\sqrt{3} and 3\sqrt{3} to get 8\sqrt{3}.

$$-\frac{5+8\sqrt{3}+3\left(\sqrt{3}\right)^{2}}{-2}$$

The square of \sqrt{3} is 3.

$$-\frac{5+8\sqrt{3}+3\times 3}{-2}$$

Multiply 3 and 3 to get 9.

$$-\frac{5+8\sqrt{3}+9}{-2}$$

Add 5 and 9 to get 14.

$$-\frac{14+8\sqrt{3}}{-2}$$

Divide each term of 14+8\sqrt{3} by -2 to get -7-4\sqrt{3}.

$$-\left(-7-4\sqrt{3}\right)$$

To find the opposite of -7-4\sqrt{3}, find the opposite of each term.

$$-\left(-7\right)-\left(-4\sqrt{3}\right)$$

The opposite of -7 is 7.

$$7-\left(-4\sqrt{3}\right)$$

The opposite of -4\sqrt{3} is 4\sqrt{3}.

$$7+4\sqrt{3}$$