Solve (4a^3-a^2/2)(a/2+4)+2a(-a^3+a)

$$( 4 a ^ { 3 } - a ^ { 2 } / 2 ) ( a / 2 + 4 ) + 2 a ( - a ^ { 3 } + a )$$

Evaluate

$$\frac{63a^{3}}{4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 4a^{3} times \frac{2}{2}.

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

Since \frac{2\times 4a^{3}}{2} and \frac{a^{2}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\times 4a^{3}-a^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{2}{2}.

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

Since \frac{a}{2} and \frac{4\times 2}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in a+4\times 2.

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

Multiply \frac{8a^{3}-a^{2}}{2} times \frac{a+8}{2} by multiplying numerator times numerator and denominator times denominator.

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

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 2a by -a^{3}+a.

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

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

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

Multiply 2 and -1 to get -2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -2a^{4}+2a^{2} times \frac{4}{4}.

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

Since \frac{\left(8a^{3}-a^{2}\right)\left(a+8\right)}{4} and \frac{4\left(-2a^{4}+2a^{2}\right)}{4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(8a^{3}-a^{2}\right)\left(a+8\right)+4\left(-2a^{4}+2a^{2}\right).

$$\frac{8a^{4}+64a^{3}-a^{3}-8a^{2}-8a^{4}+8a^{2}}{4}$$

Combine like terms in 8a^{4}+64a^{3}-a^{3}-8a^{2}-8a^{4}+8a^{2}.

$$\frac{63a^{3}}{4}$$

Expand

$$\frac{63a^{3}}{4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 4a^{3} times \frac{2}{2}.

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

Since \frac{2\times 4a^{3}}{2} and \frac{a^{2}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\times 4a^{3}-a^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{2}{2}.

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

Since \frac{a}{2} and \frac{4\times 2}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in a+4\times 2.

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

Multiply \frac{8a^{3}-a^{2}}{2} times \frac{a+8}{2} by multiplying numerator times numerator and denominator times denominator.

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

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 2a by -a^{3}+a.

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

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

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

Multiply 2 and -1 to get -2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -2a^{4}+2a^{2} times \frac{4}{4}.

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

Since \frac{\left(8a^{3}-a^{2}\right)\left(a+8\right)}{4} and \frac{4\left(-2a^{4}+2a^{2}\right)}{4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(8a^{3}-a^{2}\right)\left(a+8\right)+4\left(-2a^{4}+2a^{2}\right).

$$\frac{8a^{4}+64a^{3}-a^{3}-8a^{2}-8a^{4}+8a^{2}}{4}$$

Combine like terms in 8a^{4}+64a^{3}-a^{3}-8a^{2}-8a^{4}+8a^{2}.

$$\frac{63a^{3}}{4}$$