Solve frac{frac{4+2sqrt{3}}{12+6sqrt{2}}}{frac{2-sqrt{2}}{4-2sqrt{3}}}

$$\frac{ \frac{ 4+2 \sqrt{ 3 } }{ 12+6 \sqrt{ 2 } } }{ \frac{ 2- \sqrt{ 2 } }{ 4-2 \sqrt{ 3 } } }$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Solution Steps

Divide \frac{4+2\sqrt{3}}{12+6\sqrt{2}} by \frac{2-\sqrt{2}}{4-2\sqrt{3}} by multiplying \frac{4+2\sqrt{3}}{12+6\sqrt{2}} by the reciprocal of \frac{2-\sqrt{2}}{4-2\sqrt{3}}.

$$\frac{\left(4+2\sqrt{3}\right)\left(4-2\sqrt{3}\right)}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

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}.

$$\frac{4^{2}-\left(2\sqrt{3}\right)^{2}}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16-\left(2\sqrt{3}\right)^{2}}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

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

$$\frac{16-2^{2}\left(\sqrt{3}\right)^{2}}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{16-4\left(\sqrt{3}\right)^{2}}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{16-4\times 3}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

Multiply 4 and 3 to get 12.

$$\frac{16-12}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

Subtract 12 from 16 to get 4.

$$\frac{4}{\left(12+6\sqrt{2}\right)\left(2-\sqrt{2}\right)}$$

Apply the distributive property by multiplying each term of 12+6\sqrt{2} by each term of 2-\sqrt{2}.

$$\frac{4}{24-12\sqrt{2}+12\sqrt{2}-6\left(\sqrt{2}\right)^{2}}$$

Combine -12\sqrt{2} and 12\sqrt{2} to get 0.

$$\frac{4}{24-6\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4}{24-6\times 2}$$

Multiply -6 and 2 to get -12.

$$\frac{4}{24-12}$$

Subtract 12 from 24 to get 12.

$$\frac{4}{12}$$

Reduce the fraction \frac{4}{12} to lowest terms by extracting and canceling out 4.

$$\frac{1}{3}$$

Factor

$$\frac{1}{3} = 0.3333333333333333$$