Solve 1/2(5sqrt{3}-1/2)*3^2+(5sqrt{3}-3/2)*3

$$\frac { 1 } { 2 } ( 5 \sqrt { 3 } - \frac { 1 } { 2 } ) \times 3 ^ { 2 } + ( 5 \sqrt { 3 } - \frac { 3 } { 2 } ) \times 3$$

Evaluate

$$\frac{75\sqrt{3}}{2}-\frac{27}{4}\approx 58.201905284$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{2}\left(5\sqrt{3}-\frac{1}{2}\right)\times 9+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Multiply \frac{1}{2} and 9 to get \frac{9}{2}.

$$\frac{9}{2}\left(5\sqrt{3}-\frac{1}{2}\right)+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Use the distributive property to multiply \frac{9}{2} by 5\sqrt{3}-\frac{1}{2}.

$$\frac{9}{2}\times 5\sqrt{3}+\frac{9}{2}\left(-\frac{1}{2}\right)+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Express \frac{9}{2}\times 5 as a single fraction.

$$\frac{9\times 5}{2}\sqrt{3}+\frac{9}{2}\left(-\frac{1}{2}\right)+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Multiply 9 and 5 to get 45.

$$\frac{45}{2}\sqrt{3}+\frac{9}{2}\left(-\frac{1}{2}\right)+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Multiply \frac{9}{2} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{45}{2}\sqrt{3}+\frac{9\left(-1\right)}{2\times 2}+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Do the multiplications in the fraction \frac{9\left(-1\right)}{2\times 2}.

$$\frac{45}{2}\sqrt{3}+\frac{-9}{4}+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Fraction \frac{-9}{4} can be rewritten as -\frac{9}{4} by extracting the negative sign.

$$\frac{45}{2}\sqrt{3}-\frac{9}{4}+3\left(5\sqrt{3}-\frac{3}{2}\right)$$

Use the distributive property to multiply 3 by 5\sqrt{3}-\frac{3}{2}.

$$\frac{45}{2}\sqrt{3}-\frac{9}{4}+15\sqrt{3}+3\left(-\frac{3}{2}\right)$$

Express 3\left(-\frac{3}{2}\right) as a single fraction.

$$\frac{45}{2}\sqrt{3}-\frac{9}{4}+15\sqrt{3}+\frac{3\left(-3\right)}{2}$$

Multiply 3 and -3 to get -9.

$$\frac{45}{2}\sqrt{3}-\frac{9}{4}+15\sqrt{3}+\frac{-9}{2}$$

Fraction \frac{-9}{2} can be rewritten as -\frac{9}{2} by extracting the negative sign.

$$\frac{45}{2}\sqrt{3}-\frac{9}{4}+15\sqrt{3}-\frac{9}{2}$$

Combine \frac{45}{2}\sqrt{3} and 15\sqrt{3} to get \frac{75}{2}\sqrt{3}.

$$\frac{75}{2}\sqrt{3}-\frac{9}{4}-\frac{9}{2}$$

Least common multiple of 4 and 2 is 4. Convert -\frac{9}{4} and \frac{9}{2} to fractions with denominator 4.

$$\frac{75}{2}\sqrt{3}-\frac{9}{4}-\frac{18}{4}$$

Since -\frac{9}{4} and \frac{18}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{75}{2}\sqrt{3}+\frac{-9-18}{4}$$

Subtract 18 from -9 to get -27.

$$\frac{75}{2}\sqrt{3}-\frac{27}{4}$$

Factor

$$\frac{3 {(50 \sqrt{3} - 9)}}{4} = 58.2019052838329$$