Solve 12/(4-2sqrt{3)}*frac{4+2sqrt{3}}{(4+2sqrt{3})}

$$\frac { 12 } { ( 4 - 2 \sqrt { 3 } ) } \times \frac { 4 + 2 \sqrt { 3 } } { ( 4 + 2 \sqrt { 3 } ) }$$

Evaluate

$$6\left(\sqrt{3}+2\right)\approx 22.392304845$$

Solution Steps

Divide 4+2\sqrt{3} by 4+2\sqrt{3} to get 1.

$$\frac{12}{4-2\sqrt{3}}\times 1$$

Rationalize the denominator of \frac{12}{4-2\sqrt{3}} by multiplying numerator and denominator by 4+2\sqrt{3}.

$$\frac{12\left(4+2\sqrt{3}\right)}{\left(4-2\sqrt{3}\right)\left(4+2\sqrt{3}\right)}\times 1$$

Consider \left(4-2\sqrt{3}\right)\left(4+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}.

$$\frac{12\left(4+2\sqrt{3}\right)}{4^{2}-\left(-2\sqrt{3}\right)^{2}}\times 1$$

Calculate 4 to the power of 2 and get 16.

$$\frac{12\left(4+2\sqrt{3}\right)}{16-\left(-2\sqrt{3}\right)^{2}}\times 1$$

Expand \left(-2\sqrt{3}\right)^{2}.

$$\frac{12\left(4+2\sqrt{3}\right)}{16-\left(-2\right)^{2}\left(\sqrt{3}\right)^{2}}\times 1$$

Calculate -2 to the power of 2 and get 4.

$$\frac{12\left(4+2\sqrt{3}\right)}{16-4\left(\sqrt{3}\right)^{2}}\times 1$$

The square of \sqrt{3} is 3.

$$\frac{12\left(4+2\sqrt{3}\right)}{16-4\times 3}\times 1$$

Multiply 4 and 3 to get 12.

$$\frac{12\left(4+2\sqrt{3}\right)}{16-12}\times 1$$

Subtract 12 from 16 to get 4.

$$\frac{12\left(4+2\sqrt{3}\right)}{4}\times 1$$

Divide 12\left(4+2\sqrt{3}\right) by 4 to get 3\left(4+2\sqrt{3}\right).

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

Use the distributive property to multiply 3 by 4+2\sqrt{3}.

$$\left(12+6\sqrt{3}\right)\times 1$$

Use the distributive property to multiply 12+6\sqrt{3} by 1.

$$12+6\sqrt{3}$$

Factor

$$6 {(\sqrt{3} + 2)} = 22.392304845$$