Solve [(-1/2a^{3}b)(+8a^{3}b^{2})]^{2}:(2a^{3}b^{2})^{3}+[(a^{2}b)(ab^{2})^{3}]:(-a^{2}b^{3})^2-(-3a^2)^3:(3a^3)-3ab

$$[ ( - \frac { 1 } { 2 } a ^ { 3 } b ) ( + 8 a ^ { 3 } b ^ { 2 } ) ] ^ { 2 } : ( 2 a ^ { 3 } b ^ { 2 } ) ^ { 3 } + [ ( a ^ { 2 } b ) ( a b ^ { 2 } ) ^ { 3 } ] : ( - a ^ { 2 } b ^ { 3 } ) ^ { 2 } - ( - 3 a ^ { 2 } ) ^ { 3 } : ( 3 a ^ { 3 } ) - 3 a b$$

Evaluate

$$a\left(11a^{2}-2b\right)$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{\left(-\frac{1}{2}a^{6}b\times 8b^{2}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

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

$$\frac{\left(-\frac{1}{2}a^{6}b^{3}\times 8\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

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

$$\frac{\left(-4a^{6}b^{3}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Expand \left(-4a^{6}b^{3}\right)^{2}.

$$\frac{\left(-4\right)^{2}\left(a^{6}\right)^{2}\left(b^{3}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{\left(-4\right)^{2}a^{12}\left(b^{3}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\left(-4\right)^{2}a^{12}b^{6}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Calculate -4 to the power of 2 and get 16.

$$\frac{16a^{12}b^{6}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Expand \left(2a^{3}b^{2}\right)^{3}.

$$\frac{16a^{12}b^{6}}{2^{3}\left(a^{3}\right)^{3}\left(b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{16a^{12}b^{6}}{2^{3}a^{9}\left(b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{16a^{12}b^{6}}{2^{3}a^{9}b^{6}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Calculate 2 to the power of 3 and get 8.

$$\frac{16a^{12}b^{6}}{8a^{9}b^{6}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Cancel out 8b^{6}a^{9} in both numerator and denominator.

$$2a^{3}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

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

$$2a^{3}+\frac{a^{2}ba^{3}\left(b^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$2a^{3}+\frac{a^{2}ba^{3}b^{6}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$2a^{3}+\frac{a^{5}bb^{6}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Expand \left(\left(-a^{2}\right)b^{3}\right)^{2}.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(-a^{2}\right)^{2}\left(b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(-a^{2}\right)^{2}b^{6}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Calculate -a^{2} to the power of 2 and get \left(a^{2}\right)^{2}.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(a^{2}\right)^{2}b^{6}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Cancel out b^{6} in both numerator and denominator.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-\frac{\left(-3\right)^{3}a^{6}}{3a^{3}}-3ab$$

Calculate -3 to the power of 3 and get -27.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-\frac{-27a^{6}}{3a^{3}}-3ab$$

Cancel out 3a^{3} in both numerator and denominator.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-\left(-9a^{3}\right)-3ab$$

The opposite of -9a^{3} is 9a^{3}.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}+9a^{3}-3ab$$

Combine 2a^{3} and 9a^{3} to get 11a^{3}.

$$11a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-3ab$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 11a^{3}-3ab times \frac{\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}.

$$\frac{\left(11a^{3}-3ab\right)\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}$$

Since \frac{\left(11a^{3}-3ab\right)\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}} and \frac{ba^{5}}{\left(a^{2}\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(11a^{3}-3ab\right)\left(a^{2}\right)^{2}+ba^{5}}{\left(a^{2}\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3ab times \frac{\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}.

$$11a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}+\frac{-3ab\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}$$

Since \frac{ba^{5}}{\left(a^{2}\right)^{2}} and \frac{-3ab\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}} have the same denominator, add them by adding their numerators.

$$11a^{3}+\frac{ba^{5}-3ab\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}$$

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

$$11a^{3}+\frac{ba^{5}}{a^{4}}-3ab$$

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

$$11a^{3}+ab-3ab$$

Combine ab and -3ab to get -2ab.

$$11a^{3}-2ab$$

Expand

$$11a^{3}-2ab$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{\left(-\frac{1}{2}a^{6}b\times 8b^{2}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

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

$$\frac{\left(-\frac{1}{2}a^{6}b^{3}\times 8\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

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

$$\frac{\left(-4a^{6}b^{3}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Expand \left(-4a^{6}b^{3}\right)^{2}.

$$\frac{\left(-4\right)^{2}\left(a^{6}\right)^{2}\left(b^{3}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{\left(-4\right)^{2}a^{12}\left(b^{3}\right)^{2}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\left(-4\right)^{2}a^{12}b^{6}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Calculate -4 to the power of 2 and get 16.

$$\frac{16a^{12}b^{6}}{\left(2a^{3}b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Expand \left(2a^{3}b^{2}\right)^{3}.

$$\frac{16a^{12}b^{6}}{2^{3}\left(a^{3}\right)^{3}\left(b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{16a^{12}b^{6}}{2^{3}a^{9}\left(b^{2}\right)^{3}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{16a^{12}b^{6}}{2^{3}a^{9}b^{6}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Calculate 2 to the power of 3 and get 8.

$$\frac{16a^{12}b^{6}}{8a^{9}b^{6}}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Cancel out 8b^{6}a^{9} in both numerator and denominator.

$$2a^{3}+\frac{a^{2}b\left(ab^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

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

$$2a^{3}+\frac{a^{2}ba^{3}\left(b^{2}\right)^{3}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$2a^{3}+\frac{a^{2}ba^{3}b^{6}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$2a^{3}+\frac{a^{5}bb^{6}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(\left(-a^{2}\right)b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Expand \left(\left(-a^{2}\right)b^{3}\right)^{2}.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(-a^{2}\right)^{2}\left(b^{3}\right)^{2}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(-a^{2}\right)^{2}b^{6}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Calculate -a^{2} to the power of 2 and get \left(a^{2}\right)^{2}.

$$2a^{3}+\frac{a^{5}b^{7}}{\left(a^{2}\right)^{2}b^{6}}-\frac{\left(-3a^{2}\right)^{3}}{3a^{3}}-3ab$$

Cancel out b^{6} in both numerator and denominator.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-\frac{\left(-3\right)^{3}a^{6}}{3a^{3}}-3ab$$

Calculate -3 to the power of 3 and get -27.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-\frac{-27a^{6}}{3a^{3}}-3ab$$

Cancel out 3a^{3} in both numerator and denominator.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-\left(-9a^{3}\right)-3ab$$

The opposite of -9a^{3} is 9a^{3}.

$$2a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}+9a^{3}-3ab$$

Combine 2a^{3} and 9a^{3} to get 11a^{3}.

$$11a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}-3ab$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 11a^{3}-3ab times \frac{\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}.

$$\frac{\left(11a^{3}-3ab\right)\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}$$

Since \frac{\left(11a^{3}-3ab\right)\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}} and \frac{ba^{5}}{\left(a^{2}\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(11a^{3}-3ab\right)\left(a^{2}\right)^{2}+ba^{5}}{\left(a^{2}\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3ab times \frac{\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}.

$$11a^{3}+\frac{ba^{5}}{\left(a^{2}\right)^{2}}+\frac{-3ab\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}$$

Since \frac{ba^{5}}{\left(a^{2}\right)^{2}} and \frac{-3ab\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}} have the same denominator, add them by adding their numerators.

$$11a^{3}+\frac{ba^{5}-3ab\left(a^{2}\right)^{2}}{\left(a^{2}\right)^{2}}$$

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

$$11a^{3}+\frac{ba^{5}}{a^{4}}-3ab$$

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

$$11a^{3}+ab-3ab$$

Combine ab and -3ab to get -2ab.

$$11a^{3}-2ab$$