Solve [4(-1/2a^{2}b)^{3}]^{2}+5(-1/2a^{4}b^{2})^{3}+19(-1/2a^{3}b^2)^2(1/2a^6b^2)

$$[ 4 ( - \frac { 1 } { 2 } a ^ { 2 } b ) ^ { 3 } ] ^ { 2 } + 5 ( - \frac { 1 } { 2 } a ^ { 4 } b ^ { 2 } ) ^ { 3 } + 19 ( - \frac { 1 } { 2 } a ^ { 3 } b ^ { 2 } ) ^ { 2 } ( \frac { 1 } { 2 } a ^ { 6 } b ^ { 2 } )$$

Evaluate

$$2b^{6}a^{12}$$

Solution Steps

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

$$\left(4\left(-\frac{1}{2}\right)^{3}\left(a^{2}\right)^{3}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(4\left(-\frac{1}{2}\right)^{3}a^{6}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(4\left(-\frac{1}{8}\right)a^{6}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(-\frac{1}{2}a^{6}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Expand \left(-\frac{1}{2}a^{6}b^{3}\right)^{2}.

$$\left(-\frac{1}{2}\right)^{2}\left(a^{6}\right)^{2}\left(b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(-\frac{1}{2}\right)^{2}a^{12}\left(b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(-\frac{1}{2}\right)^{2}a^{12}b^{6}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Calculate -\frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Expand \left(-\frac{1}{2}a^{4}b^{2}\right)^{3}.

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}\right)^{3}\left(a^{4}\right)^{3}\left(b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}\right)^{3}a^{12}\left(b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}\right)^{3}a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{8}\right)a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Multiply 5 and -\frac{1}{8} to get -\frac{5}{8}.

$$\frac{1}{4}a^{12}b^{6}-\frac{5}{8}a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Combine \frac{1}{4}a^{12}b^{6} and -\frac{5}{8}a^{12}b^{6} to get -\frac{3}{8}a^{12}b^{6}.

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}\right)^{2}\left(a^{3}\right)^{2}\left(b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}\right)^{2}a^{6}\left(b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}\right)^{2}a^{6}b^{4}\times \frac{1}{2}a^{6}b^{2}$$

Calculate -\frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$-\frac{3}{8}a^{12}b^{6}+19\times \frac{1}{4}a^{6}b^{4}\times \frac{1}{2}a^{6}b^{2}$$

Multiply 19 and \frac{1}{4} to get \frac{19}{4}.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{4}a^{6}b^{4}\times \frac{1}{2}a^{6}b^{2}$$

Multiply \frac{19}{4} and \frac{1}{2} to get \frac{19}{8}.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{8}a^{6}b^{4}a^{6}b^{2}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{8}a^{12}b^{4}b^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{8}a^{12}b^{6}$$

Combine -\frac{3}{8}a^{12}b^{6} and \frac{19}{8}a^{12}b^{6} to get 2a^{12}b^{6}.

$$2a^{12}b^{6}$$

Expand

$$2b^{6}a^{12}$$

Solution Steps

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

$$\left(4\left(-\frac{1}{2}\right)^{3}\left(a^{2}\right)^{3}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(4\left(-\frac{1}{2}\right)^{3}a^{6}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(4\left(-\frac{1}{8}\right)a^{6}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(-\frac{1}{2}a^{6}b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Expand \left(-\frac{1}{2}a^{6}b^{3}\right)^{2}.

$$\left(-\frac{1}{2}\right)^{2}\left(a^{6}\right)^{2}\left(b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(-\frac{1}{2}\right)^{2}a^{12}\left(b^{3}\right)^{2}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\left(-\frac{1}{2}\right)^{2}a^{12}b^{6}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Calculate -\frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}a^{4}b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Expand \left(-\frac{1}{2}a^{4}b^{2}\right)^{3}.

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}\right)^{3}\left(a^{4}\right)^{3}\left(b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}\right)^{3}a^{12}\left(b^{2}\right)^{3}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{2}\right)^{3}a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$\frac{1}{4}a^{12}b^{6}+5\left(-\frac{1}{8}\right)a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Multiply 5 and -\frac{1}{8} to get -\frac{5}{8}.

$$\frac{1}{4}a^{12}b^{6}-\frac{5}{8}a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

Combine \frac{1}{4}a^{12}b^{6} and -\frac{5}{8}a^{12}b^{6} to get -\frac{3}{8}a^{12}b^{6}.

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}a^{3}b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}\right)^{2}\left(a^{3}\right)^{2}\left(b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}\right)^{2}a^{6}\left(b^{2}\right)^{2}\times \frac{1}{2}a^{6}b^{2}$$

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

$$-\frac{3}{8}a^{12}b^{6}+19\left(-\frac{1}{2}\right)^{2}a^{6}b^{4}\times \frac{1}{2}a^{6}b^{2}$$

Calculate -\frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$-\frac{3}{8}a^{12}b^{6}+19\times \frac{1}{4}a^{6}b^{4}\times \frac{1}{2}a^{6}b^{2}$$

Multiply 19 and \frac{1}{4} to get \frac{19}{4}.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{4}a^{6}b^{4}\times \frac{1}{2}a^{6}b^{2}$$

Multiply \frac{19}{4} and \frac{1}{2} to get \frac{19}{8}.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{8}a^{6}b^{4}a^{6}b^{2}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{8}a^{12}b^{4}b^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$-\frac{3}{8}a^{12}b^{6}+\frac{19}{8}a^{12}b^{6}$$

Combine -\frac{3}{8}a^{12}b^{6} and \frac{19}{8}a^{12}b^{6} to get 2a^{12}b^{6}.

$$2a^{12}b^{6}$$