Solve frac{3+8sqrt{3}}{12+4sqrt{3}}

$$\frac { 3 + 8 \sqrt { 3 } } { 12 + 4 \sqrt { 3 } }$$

Evaluate

$$\frac{7\sqrt{3}-5}{8}\approx 0.890544457$$

Solution Steps

Rationalize the denominator of \frac{3+8\sqrt{3}}{12+4\sqrt{3}} by multiplying numerator and denominator by 12-4\sqrt{3}.

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{\left(12+4\sqrt{3}\right)\left(12-4\sqrt{3}\right)}$$

Consider \left(12+4\sqrt{3}\right)\left(12-4\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}.

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{12^{2}-\left(4\sqrt{3}\right)^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-\left(4\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-4^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-16\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-16\times 3}$$

Multiply 16 and 3 to get 48.

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-48}$$

Subtract 48 from 144 to get 96.

$$\frac{\left(3+8\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{96}$$

Apply the distributive property by multiplying each term of 3+8\sqrt{3} by each term of 12-4\sqrt{3}.

$$\frac{36-12\sqrt{3}+96\sqrt{3}-32\left(\sqrt{3}\right)^{2}}{96}$$

Combine -12\sqrt{3} and 96\sqrt{3} to get 84\sqrt{3}.

$$\frac{36+84\sqrt{3}-32\left(\sqrt{3}\right)^{2}}{96}$$

The square of \sqrt{3} is 3.

$$\frac{36+84\sqrt{3}-32\times 3}{96}$$

Multiply -32 and 3 to get -96.

$$\frac{36+84\sqrt{3}-96}{96}$$

Subtract 96 from 36 to get -60.

$$\frac{-60+84\sqrt{3}}{96}$$

Factor

$$\frac{7 \sqrt{3} - 5}{8} = 0.8905444566227676$$