Solve 12left(120-175right)/12+frac{2*10{sqrt{3}}}

$$\frac{ 12 \left( 120-175 \right) }{ 12+ \frac{ 2 \cdot 10 }{ \sqrt{ 3 } } }$$

Evaluate

$$\frac{825\sqrt{3}-1485}{2}\approx -28.029041878$$

Short Solution Steps

Subtract 175 from 120 to get -55.

$$\frac{12\left(-55\right)}{12+\frac{2\times 10}{\sqrt{3}}}$$

Multiply 12 and -55 to get -660.

$$\frac{-660}{12+\frac{2\times 10}{\sqrt{3}}}$$

Multiply 2 and 10 to get 20.

$$\frac{-660}{12+\frac{20}{\sqrt{3}}}$$

Rationalize the denominator of \frac{20}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{-660}{12+\frac{20\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{-660}{12+\frac{20\sqrt{3}}{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12 times \frac{3}{3}.

$$\frac{-660}{\frac{12\times 3}{3}+\frac{20\sqrt{3}}{3}}$$

Since \frac{12\times 3}{3} and \frac{20\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{-660}{\frac{12\times 3+20\sqrt{3}}{3}}$$

Do the multiplications in 12\times 3+20\sqrt{3}.

$$\frac{-660}{\frac{36+20\sqrt{3}}{3}}$$

Divide -660 by \frac{36+20\sqrt{3}}{3} by multiplying -660 by the reciprocal of \frac{36+20\sqrt{3}}{3}.

$$\frac{-660\times 3}{36+20\sqrt{3}}$$

Rationalize the denominator of \frac{-660\times 3}{36+20\sqrt{3}} by multiplying numerator and denominator by 36-20\sqrt{3}.

$$\frac{-660\times 3\left(36-20\sqrt{3}\right)}{\left(36+20\sqrt{3}\right)\left(36-20\sqrt{3}\right)}$$

Consider \left(36+20\sqrt{3}\right)\left(36-20\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}.

$$\frac{-660\times 3\left(36-20\sqrt{3}\right)}{36^{2}-\left(20\sqrt{3}\right)^{2}}$$

Multiply -660 and 3 to get -1980.

$$\frac{-1980\left(36-20\sqrt{3}\right)}{36^{2}-\left(20\sqrt{3}\right)^{2}}$$

Calculate 36 to the power of 2 and get 1296.

$$\frac{-1980\left(36-20\sqrt{3}\right)}{1296-\left(20\sqrt{3}\right)^{2}}$$

Expand \left(20\sqrt{3}\right)^{2}.

$$\frac{-1980\left(36-20\sqrt{3}\right)}{1296-20^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{-1980\left(36-20\sqrt{3}\right)}{1296-400\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{-1980\left(36-20\sqrt{3}\right)}{1296-400\times 3}$$

Multiply 400 and 3 to get 1200.

$$\frac{-1980\left(36-20\sqrt{3}\right)}{1296-1200}$$

Subtract 1200 from 1296 to get 96.

$$\frac{-1980\left(36-20\sqrt{3}\right)}{96}$$

Divide -1980\left(36-20\sqrt{3}\right) by 96 to get -\frac{165}{8}\left(36-20\sqrt{3}\right).

$$-\frac{165}{8}\left(36-20\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{165}{8} by 36-20\sqrt{3}.

$$-\frac{165}{8}\times 36-\frac{165}{8}\left(-20\right)\sqrt{3}$$

Express -\frac{165}{8}\times 36 as a single fraction.

$$\frac{-165\times 36}{8}-\frac{165}{8}\left(-20\right)\sqrt{3}$$

Multiply -165 and 36 to get -5940.

$$\frac{-5940}{8}-\frac{165}{8}\left(-20\right)\sqrt{3}$$

Reduce the fraction \frac{-5940}{8} to lowest terms by extracting and canceling out 4.

$$-\frac{1485}{2}-\frac{165}{8}\left(-20\right)\sqrt{3}$$

Express -\frac{165}{8}\left(-20\right) as a single fraction.

$$-\frac{1485}{2}+\frac{-165\left(-20\right)}{8}\sqrt{3}$$

Multiply -165 and -20 to get 3300.

$$-\frac{1485}{2}+\frac{3300}{8}\sqrt{3}$$

Reduce the fraction \frac{3300}{8} to lowest terms by extracting and canceling out 4.

$$-\frac{1485}{2}+\frac{825}{2}\sqrt{3}$$

Factor

$$\frac{165 {(5 \sqrt{3} - 9)}}{2} = -28.029041877838196$$