Solve -6^2*2:(2+2*sqrt{27})=

$$- 6 ^ { 2 } \cdot 2 : ( 2 + 2 \cdot \sqrt { 27 } ) =$$

Evaluate

$$\frac{18-54\sqrt{3}}{13}\approx -5.810057201$$

Solution Steps

Calculate 6 to the power of 2 and get 36.

$$\frac{-36\times 2}{2+2\sqrt{27}}$$

Multiply -36 and 2 to get -72.

$$\frac{-72}{2+2\sqrt{27}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{-72}{2+2\times 3\sqrt{3}}$$

Multiply 2 and 3 to get 6.

$$\frac{-72}{2+6\sqrt{3}}$$

Rationalize the denominator of \frac{-72}{2+6\sqrt{3}} by multiplying numerator and denominator by 2-6\sqrt{3}.

$$\frac{-72\left(2-6\sqrt{3}\right)}{\left(2+6\sqrt{3}\right)\left(2-6\sqrt{3}\right)}$$

Consider \left(2+6\sqrt{3}\right)\left(2-6\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}.

$$\frac{-72\left(2-6\sqrt{3}\right)}{2^{2}-\left(6\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-72\left(2-6\sqrt{3}\right)}{4-\left(6\sqrt{3}\right)^{2}}$$

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

$$\frac{-72\left(2-6\sqrt{3}\right)}{4-6^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{-72\left(2-6\sqrt{3}\right)}{4-36\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{-72\left(2-6\sqrt{3}\right)}{4-36\times 3}$$

Multiply 36 and 3 to get 108.

$$\frac{-72\left(2-6\sqrt{3}\right)}{4-108}$$

Subtract 108 from 4 to get -104.

$$\frac{-72\left(2-6\sqrt{3}\right)}{-104}$$

Divide -72\left(2-6\sqrt{3}\right) by -104 to get \frac{9}{13}\left(2-6\sqrt{3}\right).

$$\frac{9}{13}\left(2-6\sqrt{3}\right)$$

Use the distributive property to multiply \frac{9}{13} by 2-6\sqrt{3}.

$$\frac{9}{13}\times 2+\frac{9}{13}\left(-6\right)\sqrt{3}$$

Express \frac{9}{13}\times 2 as a single fraction.

$$\frac{9\times 2}{13}+\frac{9}{13}\left(-6\right)\sqrt{3}$$

Multiply 9 and 2 to get 18.

$$\frac{18}{13}+\frac{9}{13}\left(-6\right)\sqrt{3}$$

Express \frac{9}{13}\left(-6\right) as a single fraction.

$$\frac{18}{13}+\frac{9\left(-6\right)}{13}\sqrt{3}$$

Multiply 9 and -6 to get -54.

$$\frac{18}{13}+\frac{-54}{13}\sqrt{3}$$

Fraction \frac{-54}{13} can be rewritten as -\frac{54}{13} by extracting the negative sign.

$$\frac{18}{13}-\frac{54}{13}\sqrt{3}$$