Solve (frac{3+3sqrt{3}}{2})(frac{33sqrt{3}-1}{2})(frac{232sqrt{3}-sqrt{3}}{2})

$$( \frac { 3 + 3 \sqrt { 3 } } { 2 } ) ( \frac { 33 \sqrt { 3 } - 1 } { 2 } ) ( \frac { 232 \sqrt { 3 } - \sqrt { 3 } } { 2 } )$$

Evaluate

$$\frac{33957\sqrt{3}}{4}+8316\approx 23019.812318154$$

Solution Steps

Combine 232\sqrt{3} and -\sqrt{3} to get 231\sqrt{3}.

$$\frac{3+3\sqrt{3}}{2}\times \frac{33\sqrt{3}-1}{2}\times \frac{231\sqrt{3}}{2}$$

Multiply \frac{3+3\sqrt{3}}{2} times \frac{33\sqrt{3}-1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(3+3\sqrt{3}\right)\left(33\sqrt{3}-1\right)}{2\times 2}\times \frac{231\sqrt{3}}{2}$$

Multiply \frac{\left(3+3\sqrt{3}\right)\left(33\sqrt{3}-1\right)}{2\times 2} times \frac{231\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(3+3\sqrt{3}\right)\left(33\sqrt{3}-1\right)\times 231\sqrt{3}}{2\times 2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(3+3\sqrt{3}\right)\left(33\sqrt{3}-1\right)\times 231\sqrt{3}}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(3+3\sqrt{3}\right)\left(33\sqrt{3}-1\right)\times 231\sqrt{3}}{8}$$

Apply the distributive property by multiplying each term of 3+3\sqrt{3} by each term of 33\sqrt{3}-1.

$$\frac{\left(99\sqrt{3}-3+99\left(\sqrt{3}\right)^{2}-3\sqrt{3}\right)\times 231\sqrt{3}}{8}$$

The square of \sqrt{3} is 3.

$$\frac{\left(99\sqrt{3}-3+99\times 3-3\sqrt{3}\right)\times 231\sqrt{3}}{8}$$

Multiply 99 and 3 to get 297.

$$\frac{\left(99\sqrt{3}-3+297-3\sqrt{3}\right)\times 231\sqrt{3}}{8}$$

Add -3 and 297 to get 294.

$$\frac{\left(99\sqrt{3}+294-3\sqrt{3}\right)\times 231\sqrt{3}}{8}$$

Combine 99\sqrt{3} and -3\sqrt{3} to get 96\sqrt{3}.

$$\frac{\left(96\sqrt{3}+294\right)\times 231\sqrt{3}}{8}$$

Use the distributive property to multiply 96\sqrt{3}+294 by 231.

$$\frac{\left(22176\sqrt{3}+67914\right)\sqrt{3}}{8}$$

Use the distributive property to multiply 22176\sqrt{3}+67914 by \sqrt{3}.

$$\frac{22176\left(\sqrt{3}\right)^{2}+67914\sqrt{3}}{8}$$

The square of \sqrt{3} is 3.

$$\frac{22176\times 3+67914\sqrt{3}}{8}$$

Multiply 22176 and 3 to get 66528.

$$\frac{66528+67914\sqrt{3}}{8}$$

Factor

$$\frac{693 {(49 \sqrt{3} + 48)}}{4} = 23019.81231815409$$