Solve 3/2a(1+3a)(3a-1)+3(1/2a+1/3)(a+1/3)

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

Evaluate

$$\frac{27a^{3}}{2}+\frac{3a^{2}}{2}+\frac{1}{3}$$

Solution Steps

Use the distributive property to multiply \frac{3}{2}a by 1+3a.

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

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

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

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

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

Multiply 3 and 3 to get 9.

$$\left(\frac{3}{2}a+\frac{9}{2}a^{2}\right)\left(3a-1\right)+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Apply the distributive property by multiplying each term of \frac{3}{2}a+\frac{9}{2}a^{2} by each term of 3a-1.

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

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

Multiply 3 and 3 to get 9.

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

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

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

Express \frac{9}{2}\times 3 as a single fraction.

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

Multiply 9 and 3 to get 27.

$$\frac{9}{2}a^{2}-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{9}{2}a^{2}\left(-1\right)+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

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

$$\frac{9}{2}a^{2}-\frac{3}{2}a+\frac{27}{2}a^{3}-\frac{9}{2}a^{2}+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Combine \frac{9}{2}a^{2} and -\frac{9}{2}a^{2} to get 0.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Use the distributive property to multiply 3 by \frac{1}{2}a+\frac{1}{3}.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\left(3\times \frac{1}{2}a+3\times \frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Multiply 3 and \frac{1}{2} to get \frac{3}{2}.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\left(\frac{3}{2}a+3\times \frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Cancel out 3 and 3.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\left(\frac{3}{2}a+1\right)\left(a+\frac{1}{3}\right)$$

Apply the distributive property by multiplying each term of \frac{3}{2}a+1 by each term of a+\frac{1}{3}.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}aa+\frac{3}{2}a\times \frac{1}{3}+a+\frac{1}{3}$$

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

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{3}{2}a\times \frac{1}{3}+a+\frac{1}{3}$$

Multiply \frac{3}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{3\times 1}{2\times 3}a+a+\frac{1}{3}$$

Cancel out 3 in both numerator and denominator.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a+a+\frac{1}{3}$$

Combine \frac{1}{2}a and a to get \frac{3}{2}a.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{3}{2}a+\frac{1}{3}$$

Combine -\frac{3}{2}a and \frac{3}{2}a to get 0.

$$\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{1}{3}$$

Expand

$$\frac{27a^{3}}{2}+\frac{3a^{2}}{2}+\frac{1}{3}$$

Solution Steps

Use the distributive property to multiply \frac{3}{2}a by 1+3a.

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

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

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

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

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

Multiply 3 and 3 to get 9.

$$\left(\frac{3}{2}a+\frac{9}{2}a^{2}\right)\left(3a-1\right)+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Apply the distributive property by multiplying each term of \frac{3}{2}a+\frac{9}{2}a^{2} by each term of 3a-1.

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

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

Multiply 3 and 3 to get 9.

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

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

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

Express \frac{9}{2}\times 3 as a single fraction.

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

Multiply 9 and 3 to get 27.

$$\frac{9}{2}a^{2}-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{9}{2}a^{2}\left(-1\right)+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

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

$$\frac{9}{2}a^{2}-\frac{3}{2}a+\frac{27}{2}a^{3}-\frac{9}{2}a^{2}+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Combine \frac{9}{2}a^{2} and -\frac{9}{2}a^{2} to get 0.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+3\left(\frac{1}{2}a+\frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Use the distributive property to multiply 3 by \frac{1}{2}a+\frac{1}{3}.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\left(3\times \frac{1}{2}a+3\times \frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Multiply 3 and \frac{1}{2} to get \frac{3}{2}.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\left(\frac{3}{2}a+3\times \frac{1}{3}\right)\left(a+\frac{1}{3}\right)$$

Cancel out 3 and 3.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\left(\frac{3}{2}a+1\right)\left(a+\frac{1}{3}\right)$$

Apply the distributive property by multiplying each term of \frac{3}{2}a+1 by each term of a+\frac{1}{3}.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}aa+\frac{3}{2}a\times \frac{1}{3}+a+\frac{1}{3}$$

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

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{3}{2}a\times \frac{1}{3}+a+\frac{1}{3}$$

Multiply \frac{3}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{3\times 1}{2\times 3}a+a+\frac{1}{3}$$

Cancel out 3 in both numerator and denominator.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a+a+\frac{1}{3}$$

Combine \frac{1}{2}a and a to get \frac{3}{2}a.

$$-\frac{3}{2}a+\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{3}{2}a+\frac{1}{3}$$

Combine -\frac{3}{2}a and \frac{3}{2}a to get 0.

$$\frac{27}{2}a^{3}+\frac{3}{2}a^{2}+\frac{1}{3}$$