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

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

Evaluate

$$5\sqrt{3}+8\approx 16.660254038$$

Solution Steps

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

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

Consider \left(2-\sqrt{3}\right)\left(2+\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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

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

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

Combine \sqrt{3} and 4\sqrt{3} to get 5\sqrt{3}.

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

The square of \sqrt{3} is 3.

$$2+5\sqrt{3}+2\times 3$$

Multiply 2 and 3 to get 6.

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

Add 2 and 6 to get 8.

$$8+5\sqrt{3}$$