Solve frac{-3sqrt{3}}{(2-3sqrt{3})(2+3sqrt{3})-(3sqrt{3}-2)^2}

$$\frac { - 3 \sqrt { 3 } } { ( 2 - 3 \sqrt { 3 } ) ( 2 + 3 \sqrt { 3 } ) - ( 3 \sqrt { 3 } - 2 ) ^ { 2 } }$$

Evaluate

$$\frac{3\sqrt{3}+2}{46}\approx 0.156438096$$

Solution Steps

Consider \left(2-3\sqrt{3}\right)\left(2+3\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}. Square 2.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{3} is 3.

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

Multiply 9 and 3 to get 27.

$$\frac{-3\sqrt{3}}{4-27-\left(3\sqrt{3}-2\right)^{2}}$$

Subtract 27 from 4 to get -23.

$$\frac{-3\sqrt{3}}{-23-\left(3\sqrt{3}-2\right)^{2}}$$

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

$$\frac{-3\sqrt{3}}{-23-\left(9\left(\sqrt{3}\right)^{2}-12\sqrt{3}+4\right)}$$

The square of \sqrt{3} is 3.

$$\frac{-3\sqrt{3}}{-23-\left(9\times 3-12\sqrt{3}+4\right)}$$

Multiply 9 and 3 to get 27.

$$\frac{-3\sqrt{3}}{-23-\left(27-12\sqrt{3}+4\right)}$$

Add 27 and 4 to get 31.

$$\frac{-3\sqrt{3}}{-23-\left(31-12\sqrt{3}\right)}$$

To find the opposite of 31-12\sqrt{3}, find the opposite of each term.

$$\frac{-3\sqrt{3}}{-23-31+12\sqrt{3}}$$

Subtract 31 from -23 to get -54.

$$\frac{-3\sqrt{3}}{-54+12\sqrt{3}}$$

Rationalize the denominator of \frac{-3\sqrt{3}}{-54+12\sqrt{3}} by multiplying numerator and denominator by -54-12\sqrt{3}.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{\left(-54+12\sqrt{3}\right)\left(-54-12\sqrt{3}\right)}$$

Consider \left(-54+12\sqrt{3}\right)\left(-54-12\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}.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{\left(-54\right)^{2}-\left(12\sqrt{3}\right)^{2}}$$

Calculate -54 to the power of 2 and get 2916.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{2916-\left(12\sqrt{3}\right)^{2}}$$

Expand \left(12\sqrt{3}\right)^{2}.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{2916-12^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{2916-144\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{2916-144\times 3}$$

Multiply 144 and 3 to get 432.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{2916-432}$$

Subtract 432 from 2916 to get 2484.

$$\frac{-3\sqrt{3}\left(-54-12\sqrt{3}\right)}{2484}$$

Divide -3\sqrt{3}\left(-54-12\sqrt{3}\right) by 2484 to get -\frac{1}{828}\sqrt{3}\left(-54-12\sqrt{3}\right).

$$-\frac{1}{828}\sqrt{3}\left(-54-12\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{1}{828}\sqrt{3} by -54-12\sqrt{3}.

$$\frac{3}{46}\sqrt{3}+\frac{1}{69}\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{3}{46}\sqrt{3}+\frac{1}{69}\times 3$$

Multiply \frac{1}{69} and 3 to get \frac{1}{23}.

$$\frac{3}{46}\sqrt{3}+\frac{1}{23}$$