Solve frac{4*(6-3sqrt{3})^2+1}{12-6sqrt{3}}

$$\frac { 4 \times ( 6 - 3 \sqrt { 3 } ) ^ { 2 } + 1 } { 12 - 6 \sqrt { 3 } }$$

Evaluate

$$-\frac{35\sqrt{3}}{6}+\frac{37}{3}\approx 2.229703623$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-3\sqrt{3}\right)^{2}.

$$\frac{4\left(36-36\sqrt{3}+9\left(\sqrt{3}\right)^{2}\right)+1}{12-6\sqrt{3}}$$

The square of \sqrt{3} is 3.

$$\frac{4\left(36-36\sqrt{3}+9\times 3\right)+1}{12-6\sqrt{3}}$$

Multiply 9 and 3 to get 27.

$$\frac{4\left(36-36\sqrt{3}+27\right)+1}{12-6\sqrt{3}}$$

Add 36 and 27 to get 63.

$$\frac{4\left(63-36\sqrt{3}\right)+1}{12-6\sqrt{3}}$$

Rationalize the denominator of \frac{4\left(63-36\sqrt{3}\right)+1}{12-6\sqrt{3}} by multiplying numerator and denominator by 12+6\sqrt{3}.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{\left(12-6\sqrt{3}\right)\left(12+6\sqrt{3}\right)}$$

Consider \left(12-6\sqrt{3}\right)\left(12+6\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}.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{12^{2}-\left(-6\sqrt{3}\right)^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{144-\left(-6\sqrt{3}\right)^{2}}$$

Expand \left(-6\sqrt{3}\right)^{2}.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{144-\left(-6\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -6 to the power of 2 and get 36.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{144-36\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{144-36\times 3}$$

Multiply 36 and 3 to get 108.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{144-108}$$

Subtract 108 from 144 to get 36.

$$\frac{\left(4\left(63-36\sqrt{3}\right)+1\right)\left(12+6\sqrt{3}\right)}{36}$$

Use the distributive property to multiply 4 by 63-36\sqrt{3}.

$$\frac{\left(252-144\sqrt{3}+1\right)\left(12+6\sqrt{3}\right)}{36}$$

Add 252 and 1 to get 253.

$$\frac{\left(253-144\sqrt{3}\right)\left(12+6\sqrt{3}\right)}{36}$$

Use the distributive property to multiply 253-144\sqrt{3} by 12+6\sqrt{3} and combine like terms.

$$\frac{3036-210\sqrt{3}-864\left(\sqrt{3}\right)^{2}}{36}$$

The square of \sqrt{3} is 3.

$$\frac{3036-210\sqrt{3}-864\times 3}{36}$$

Multiply -864 and 3 to get -2592.

$$\frac{3036-210\sqrt{3}-2592}{36}$$

Subtract 2592 from 3036 to get 444.

$$\frac{444-210\sqrt{3}}{36}$$