Solve frac{sqrt{3}}{2}a*8*1/2+frac{sqrt{3}}{2}a(8-a)*1/2

$$\frac { \sqrt { 3 } } { 2 } a \times 8 \times \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } a ( 8 - a ) \times \frac { 1 } { 2 }$$

Evaluate

$$\frac{\sqrt{3}a\left(16-a\right)}{4}$$

Short Solution Steps

Multiply 8 and \frac{1}{2} to get \frac{8}{2}.

$$\frac{8}{2}\times \frac{\sqrt{3}}{2}a+\frac{\sqrt{3}}{2}a\left(8-a\right)\times \frac{1}{2}$$

Divide 8 by 2 to get 4.

$$4\times \frac{\sqrt{3}}{2}a+\frac{\sqrt{3}}{2}a\left(8-a\right)\times \frac{1}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$2\sqrt{3}a+\frac{\sqrt{3}}{2}a\left(8-a\right)\times \frac{1}{2}$$

Express \frac{\sqrt{3}}{2}a as a single fraction.

$$2\sqrt{3}a+\frac{\sqrt{3}a}{2}\left(8-a\right)\times \frac{1}{2}$$

Express \frac{\sqrt{3}a}{2}\left(8-a\right) as a single fraction.

$$2\sqrt{3}a+\frac{\sqrt{3}a\left(8-a\right)}{2}\times \frac{1}{2}$$

Multiply \frac{\sqrt{3}a\left(8-a\right)}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{3}a times \frac{2\times 2}{2\times 2}.

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

Since \frac{2\sqrt{3}a\times 2\times 2}{2\times 2} and \frac{\sqrt{3}a\left(8-a\right)}{2\times 2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\sqrt{3}a\times 2\times 2+\sqrt{3}a\left(8-a\right).

$$\frac{8\sqrt{3}a+8\sqrt{3}a-\sqrt{3}a^{2}}{2\times 2}$$

Combine like terms in 8\sqrt{3}a+8\sqrt{3}a-\sqrt{3}a^{2}.

$$\frac{16\sqrt{3}a-\sqrt{3}a^{2}}{2\times 2}$$

Expand 2\times 2.

$$\frac{16\sqrt{3}a-\sqrt{3}a^{2}}{4}$$

Factor

$$\frac{\sqrt{3}a\left(16-a\right)}{4}$$

Solution Steps

Factor out common term a by using distributive property.

$$a\left(\frac{1}{2}\times \frac{1}{2}\times 3^{\frac{1}{2}}\left(8-a\right)+\frac{1}{2}\times \frac{1}{2}\times 8\times 3^{\frac{1}{2}}\right)$$

Consider \frac{1}{2}\times \frac{1}{2}\times 3^{\frac{1}{2}}\left(8-a\right)+\frac{1}{2}\times \frac{1}{2}\times 8\times 3^{\frac{1}{2}}. Factor out \frac{1}{4}.

$$\frac{\sqrt{3}\left(8-a\right)+8\sqrt{3}}{4}$$

Consider \sqrt{3}\left(8-a\right)+8\sqrt{3}. Factor out \sqrt{3}.

$$\sqrt{3}\left(16-a\right)$$

Consider 8-a+8. Multiply and combine like terms.

$$-a+16$$

Rewrite the complete factored expression.

$$\frac{\sqrt{3}\left(-a+16\right)}{4}$$

Rewrite the complete factored expression. Simplify.

$$\frac{a\left(-a+16\right)\sqrt{3}}{4}$$