Solve 2a(-1/2a^{2}b)^{3}-ab(-a^{3}b)^{2}+(a^{2}+2a^2)^4-(-9a^4)^2

$$2 a ( - \frac { 1 } { 2 } a ^ { 2 } b ) ^ { 3 } - a b ( - a ^ { 3 } b ) ^ { 2 } + ( a ^ { 2 } + 2 a ^ { 2 } ) ^ { 4 } - ( - 9 a ^ { 4 } ) ^ { 2 }$$

Evaluate

$$-\frac{5b^{3}a^{7}}{4}$$

Solution Steps

Expand \left(-\frac{1}{2}a^{2}b\right)^{3}.

$$2a\left(-\frac{1}{2}\right)^{3}\left(a^{2}\right)^{3}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$2a\left(-\frac{1}{2}\right)^{3}a^{6}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

Calculate -\frac{1}{2} to the power of 3 and get -\frac{1}{8}.

$$2a\left(-\frac{1}{8}\right)a^{6}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}aa^{6}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab\left(-a^{3}\right)^{2}b^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

Calculate -a^{3} to the power of 2 and get \left(a^{3}\right)^{2}.

$$-\frac{1}{4}a^{7}b^{3}-ab\left(a^{3}\right)^{2}b^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+\left(3a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+3^{4}\left(a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

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

Calculate 3 to the power of 4 and get 81.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-\left(-9a^{4}\right)^{2}$$

Expand \left(-9a^{4}\right)^{2}.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-\left(-9\right)^{2}\left(a^{4}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-\left(-9\right)^{2}a^{8}$$

Calculate -9 to the power of 2 and get 81.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-81a^{8}$$

Combine 81a^{8} and -81a^{8} to get 0.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}a^{6}$$

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

$$-\frac{1}{4}a^{7}b^{3}-a^{7}b^{3}$$

Combine -\frac{1}{4}a^{7}b^{3} and -a^{7}b^{3} to get -\frac{5}{4}a^{7}b^{3}.

$$-\frac{5}{4}a^{7}b^{3}$$

Expand

$$-\frac{5b^{3}a^{7}}{4}$$

Solution Steps

Expand \left(-\frac{1}{2}a^{2}b\right)^{3}.

$$2a\left(-\frac{1}{2}\right)^{3}\left(a^{2}\right)^{3}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$2a\left(-\frac{1}{2}\right)^{3}a^{6}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

Calculate -\frac{1}{2} to the power of 3 and get -\frac{1}{8}.

$$2a\left(-\frac{1}{8}\right)a^{6}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}aa^{6}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab\left(\left(-a^{3}\right)b\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab\left(-a^{3}\right)^{2}b^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

Calculate -a^{3} to the power of 2 and get \left(a^{3}\right)^{2}.

$$-\frac{1}{4}a^{7}b^{3}-ab\left(a^{3}\right)^{2}b^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+\left(a^{2}+2a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+\left(3a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+3^{4}\left(a^{2}\right)^{4}-\left(-9a^{4}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

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

Calculate 3 to the power of 4 and get 81.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-\left(-9a^{4}\right)^{2}$$

Expand \left(-9a^{4}\right)^{2}.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-\left(-9\right)^{2}\left(a^{4}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-\left(-9\right)^{2}a^{8}$$

Calculate -9 to the power of 2 and get 81.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}+81a^{8}-81a^{8}$$

Combine 81a^{8} and -81a^{8} to get 0.

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}\left(a^{3}\right)^{2}$$

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

$$-\frac{1}{4}a^{7}b^{3}-ab^{3}a^{6}$$

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

$$-\frac{1}{4}a^{7}b^{3}-a^{7}b^{3}$$

Combine -\frac{1}{4}a^{7}b^{3} and -a^{7}b^{3} to get -\frac{5}{4}a^{7}b^{3}.

$$-\frac{5}{4}a^{7}b^{3}$$