Solve frac{12+9sqrt{3}}{12+4sqrt{3}}*frac{12-4sqrt{3}}{12-4sqrt{3}}

$$\frac { 12 + 9 \sqrt { 3 } } { 12 + 4 \sqrt { 3 } } \times \frac { 12 - 4 \sqrt { 3 } } { 12 - 4 \sqrt { 3 } }$$

Evaluate

$$\frac{5\sqrt{3}+3}{8}\approx 1.457531755$$

Solution Steps

Divide 12-4\sqrt{3} by 12-4\sqrt{3} to get 1.

$$\frac{12+9\sqrt{3}}{12+4\sqrt{3}}\times 1$$

Rationalize the denominator of \frac{12+9\sqrt{3}}{12+4\sqrt{3}} by multiplying numerator and denominator by 12-4\sqrt{3}.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{\left(12+4\sqrt{3}\right)\left(12-4\sqrt{3}\right)}\times 1$$

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}.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{12^{2}-\left(4\sqrt{3}\right)^{2}}\times 1$$

Calculate 12 to the power of 2 and get 144.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-\left(4\sqrt{3}\right)^{2}}\times 1$$

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

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-4^{2}\left(\sqrt{3}\right)^{2}}\times 1$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-16\left(\sqrt{3}\right)^{2}}\times 1$$

The square of \sqrt{3} is 3.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-16\times 3}\times 1$$

Multiply 16 and 3 to get 48.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{144-48}\times 1$$

Subtract 48 from 144 to get 96.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{96}\times 1$$

Express \frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{96}\times 1 as a single fraction.

$$\frac{\left(12+9\sqrt{3}\right)\left(12-4\sqrt{3}\right)}{96}$$

Apply the distributive property by multiplying each term of 12+9\sqrt{3} by each term of 12-4\sqrt{3}.

$$\frac{144-48\sqrt{3}+108\sqrt{3}-36\left(\sqrt{3}\right)^{2}}{96}$$

Combine -48\sqrt{3} and 108\sqrt{3} to get 60\sqrt{3}.

$$\frac{144+60\sqrt{3}-36\left(\sqrt{3}\right)^{2}}{96}$$

The square of \sqrt{3} is 3.

$$\frac{144+60\sqrt{3}-36\times 3}{96}$$

Multiply -36 and 3 to get -108.

$$\frac{144+60\sqrt{3}-108}{96}$$

Subtract 108 from 144 to get 36.

$$\frac{36+60\sqrt{3}}{96}$$

Factor

$$\frac{\sqrt{3} {(\sqrt{3} + 5)}}{8} = 1.4575317547305482$$