Solve [(-3/2abc^{2})^{2}*(-1/2a^{2}bc^3)]:(+3/2abc^2)^2+(-1/2abc^2)(-frac{1}{2}ac)

$$[ ( - \frac { 3 } { 2 } a b c ^ { 2 } ) ^ { 2 } \cdot ( - \frac { 1 } { 2 } a ^ { 2 } b c ^ { 3 } ) ] : ( + \frac { 3 } { 2 } a b c ^ { 2 } ) ^ { 2 } + ( - \frac { 1 } { 2 } a b c ^ { 2 } ) ( - \frac { 1 } { 2 } a c )$$

Evaluate

$$-\frac{ba^{2}c^{3}}{4}$$

Solution Steps

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

$$\frac{\left(-\frac{3}{2}abc^{2}\right)^{2}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{2}\left(-\frac{1}{2}\right)c$$

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

$$\frac{\left(-\frac{3}{2}abc^{2}\right)^{2}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Expand \left(-\frac{3}{2}abc^{2}\right)^{2}.

$$\frac{\left(-\frac{3}{2}\right)^{2}a^{2}b^{2}\left(c^{2}\right)^{2}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(-\frac{3}{2}\right)^{2}a^{2}b^{2}c^{4}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Calculate -\frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{\frac{9}{4}a^{2}b^{2}c^{4}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Multiply \frac{9}{4} and -\frac{1}{2} to get -\frac{9}{8}.

$$\frac{-\frac{9}{8}a^{2}b^{2}c^{4}a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{-\frac{9}{8}a^{4}b^{2}c^{4}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

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

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{4}c^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Expand \left(\frac{3}{2}abc^{2}\right)^{2}.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\left(\frac{3}{2}\right)^{2}a^{2}b^{2}\left(c^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\left(\frac{3}{2}\right)^{2}a^{2}b^{2}c^{4}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\frac{9}{4}a^{2}b^{2}c^{4}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Cancel out a^{2}b^{2}c^{4} in both numerator and denominator.

$$\frac{-\frac{9}{8}ba^{2}c^{3}}{\frac{9}{4}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Divide -\frac{9}{8}ba^{2}c^{3} by \frac{9}{4} by multiplying -\frac{9}{8}ba^{2}c^{3} by the reciprocal of \frac{9}{4}.

$$\frac{-\frac{9}{8}ba^{2}c^{3}\times 4}{9}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Multiply -\frac{9}{8} and 4 to get -\frac{9}{2}.

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

Divide -\frac{9}{2}ba^{2}c^{3} by 9 to get -\frac{1}{2}ba^{2}c^{3}.

$$-\frac{1}{2}ba^{2}c^{3}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Multiply -\frac{1}{2} and -\frac{1}{2} to get \frac{1}{4}.

$$-\frac{1}{2}ba^{2}c^{3}+\frac{1}{4}a^{2}bc^{3}$$

Combine -\frac{1}{2}ba^{2}c^{3} and \frac{1}{4}a^{2}bc^{3} to get -\frac{1}{4}ba^{2}c^{3}.

$$-\frac{1}{4}ba^{2}c^{3}$$

Expand

$$-\frac{ba^{2}c^{3}}{4}$$

Solution Steps

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

$$\frac{\left(-\frac{3}{2}abc^{2}\right)^{2}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{2}\left(-\frac{1}{2}\right)c$$

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

$$\frac{\left(-\frac{3}{2}abc^{2}\right)^{2}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Expand \left(-\frac{3}{2}abc^{2}\right)^{2}.

$$\frac{\left(-\frac{3}{2}\right)^{2}a^{2}b^{2}\left(c^{2}\right)^{2}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(-\frac{3}{2}\right)^{2}a^{2}b^{2}c^{4}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Calculate -\frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{\frac{9}{4}a^{2}b^{2}c^{4}\left(-\frac{1}{2}\right)a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Multiply \frac{9}{4} and -\frac{1}{2} to get -\frac{9}{8}.

$$\frac{-\frac{9}{8}a^{2}b^{2}c^{4}a^{2}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{-\frac{9}{8}a^{4}b^{2}c^{4}bc^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

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

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{4}c^{3}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\left(\frac{3}{2}abc^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Expand \left(\frac{3}{2}abc^{2}\right)^{2}.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\left(\frac{3}{2}\right)^{2}a^{2}b^{2}\left(c^{2}\right)^{2}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\left(\frac{3}{2}\right)^{2}a^{2}b^{2}c^{4}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{-\frac{9}{8}a^{4}b^{3}c^{7}}{\frac{9}{4}a^{2}b^{2}c^{4}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Cancel out a^{2}b^{2}c^{4} in both numerator and denominator.

$$\frac{-\frac{9}{8}ba^{2}c^{3}}{\frac{9}{4}}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Divide -\frac{9}{8}ba^{2}c^{3} by \frac{9}{4} by multiplying -\frac{9}{8}ba^{2}c^{3} by the reciprocal of \frac{9}{4}.

$$\frac{-\frac{9}{8}ba^{2}c^{3}\times 4}{9}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Multiply -\frac{9}{8} and 4 to get -\frac{9}{2}.

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

Divide -\frac{9}{2}ba^{2}c^{3} by 9 to get -\frac{1}{2}ba^{2}c^{3}.

$$-\frac{1}{2}ba^{2}c^{3}-\frac{1}{2}a^{2}bc^{3}\left(-\frac{1}{2}\right)$$

Multiply -\frac{1}{2} and -\frac{1}{2} to get \frac{1}{4}.

$$-\frac{1}{2}ba^{2}c^{3}+\frac{1}{4}a^{2}bc^{3}$$

Combine -\frac{1}{2}ba^{2}c^{3} and \frac{1}{4}a^{2}bc^{3} to get -\frac{1}{4}ba^{2}c^{3}.

$$-\frac{1}{4}ba^{2}c^{3}$$