Solve (a-1)(2a+1)+(-a)^2+(5a+1)(-a)+(-1/3a)^3:(-1/3a)^2+1

$$( a - 1 ) ( 2 a + 1 ) + ( - a ) ^ { 2 } + ( 5 a + 1 ) ( - a ) + ( - \frac { 1 } { 3 } a ) ^ { 3 } : ( - \frac { 1 } { 3 } a ) ^ { 2 } + 1$$

Evaluate

$$-\frac{a\left(6a+7\right)}{3}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 2 from 3 to get 1.

$$\left(a-1\right)\left(2a+1\right)+\left(-a\right)^{2}+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Use the distributive property to multiply a-1 by 2a+1 and combine like terms.

$$2a^{2}-a-1+\left(-a\right)^{2}+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Calculate -a to the power of 2 and get a^{2}.

$$2a^{2}-a-1+a^{2}+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Combine 2a^{2} and a^{2} to get 3a^{2}.

$$3a^{2}-a-1+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Use the distributive property to multiply 5a+1 by -a.

$$3a^{2}-a-1+5a\left(-a\right)-a+\left(-\frac{1}{3}a\right)^{1}+1$$

Calculate -\frac{1}{3}a to the power of 1 and get -\frac{1}{3}a.

$$3a^{2}-a-1+5a\left(-a\right)-a-\frac{1}{3}a+1$$

Combine -a and -\frac{1}{3}a to get -\frac{4}{3}a.

$$3a^{2}-\frac{4}{3}a-1+5a\left(-a\right)-a+1$$

Add -1 and 1 to get 0.

$$3a^{2}-\frac{4}{3}a+5a\left(-a\right)-a$$

Multiply a and a to get a^{2}.

$$3a^{2}-\frac{4}{3}a+5a^{2}\left(-1\right)-a$$

Multiply 5 and -1 to get -5.

$$3a^{2}-\frac{4}{3}a-5a^{2}-a$$

Combine 3a^{2} and -5a^{2} to get -2a^{2}.

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

Combine -\frac{4}{3}a and -a to get -\frac{7}{3}a.

$$-2a^{2}-\frac{7}{3}a$$

Expand

$$-2a^{2}-\frac{7a}{3}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 2 from 3 to get 1.

$$\left(a-1\right)\left(2a+1\right)+\left(-a\right)^{2}+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Use the distributive property to multiply a-1 by 2a+1 and combine like terms.

$$2a^{2}-a-1+\left(-a\right)^{2}+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Calculate -a to the power of 2 and get a^{2}.

$$2a^{2}-a-1+a^{2}+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Combine 2a^{2} and a^{2} to get 3a^{2}.

$$3a^{2}-a-1+\left(5a+1\right)\left(-a\right)+\left(-\frac{1}{3}a\right)^{1}+1$$

Use the distributive property to multiply 5a+1 by -a.

$$3a^{2}-a-1+5a\left(-a\right)-a+\left(-\frac{1}{3}a\right)^{1}+1$$

Calculate -\frac{1}{3}a to the power of 1 and get -\frac{1}{3}a.

$$3a^{2}-a-1+5a\left(-a\right)-a-\frac{1}{3}a+1$$

Combine -a and -\frac{1}{3}a to get -\frac{4}{3}a.

$$3a^{2}-\frac{4}{3}a-1+5a\left(-a\right)-a+1$$

Add -1 and 1 to get 0.

$$3a^{2}-\frac{4}{3}a+5a\left(-a\right)-a$$

Multiply a and a to get a^{2}.

$$3a^{2}-\frac{4}{3}a+5a^{2}\left(-1\right)-a$$

Multiply 5 and -1 to get -5.

$$3a^{2}-\frac{4}{3}a-5a^{2}-a$$

Combine 3a^{2} and -5a^{2} to get -2a^{2}.

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

Combine -\frac{4}{3}a and -a to get -\frac{7}{3}a.

$$-2a^{2}-\frac{7}{3}a$$