Solve frac{5+2sqrt{3}}{7+4sqrt{3}}

$$\frac{ 5+2 \sqrt{ 3 } }{ 7+4 \sqrt{ 3 } }$$

Evaluate

$$11-6\sqrt{3}\approx 0.607695155$$

Solution Steps

Rationalize the denominator of \frac{5+2\sqrt{3}}{7+4\sqrt{3}} by multiplying numerator and denominator by 7-4\sqrt{3}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}$$

Consider \left(7+4\sqrt{3}\right)\left(7-4\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}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{7^{2}-\left(4\sqrt{3}\right)^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-\left(4\sqrt{3}\right)^{2}}$$

Expand \left(4\sqrt{3}\right)^{2}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-4^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\times 3}$$

Multiply 16 and 3 to get 48.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-48}$$

Subtract 48 from 49 to get 1.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{1}$$

Anything divided by one gives itself.

$$\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)$$

Apply the distributive property by multiplying each term of 5+2\sqrt{3} by each term of 7-4\sqrt{3}.

$$35-20\sqrt{3}+14\sqrt{3}-8\left(\sqrt{3}\right)^{2}$$

Combine -20\sqrt{3} and 14\sqrt{3} to get -6\sqrt{3}.

$$35-6\sqrt{3}-8\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$35-6\sqrt{3}-8\times 3$$

Multiply -8 and 3 to get -24.

$$35-6\sqrt{3}-24$$

Subtract 24 from 35 to get 11.

$$11-6\sqrt{3}$$