Solve (-2b)^{4}+(-2/3b^{2}+1/2b^{2})^2*(-18b^4)+1/2(3b^4)^2

$$( - 2 b ) ^ { 4 } + ( - \frac { 2 } { 3 } b ^ { 2 } + \frac { 1 } { 2 } b ^ { 2 } ) ^ { 2 } \cdot ( - 18 b ^ { 4 } ) + \frac { 1 } { 2 } ( 3 b ^ { 4 } ) ^ { 2 }$$

Evaluate

$$4b^{4}\left(b^{4}+4\right)$$

Solution Steps

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

$$\left(-2\right)^{4}b^{4}+\left(-\frac{2}{3}b^{2}+\frac{1}{2}b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}+\left(-\frac{2}{3}b^{2}+\frac{1}{2}b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

Combine -\frac{2}{3}b^{2} and \frac{1}{2}b^{2} to get -\frac{1}{6}b^{2}.

$$16b^{4}+\left(-\frac{1}{6}b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}+\left(-\frac{1}{6}\right)^{2}\left(b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}+\left(-\frac{1}{6}\right)^{2}b^{4}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

Calculate -\frac{1}{6} to the power of 2 and get \frac{1}{36}.

$$16b^{4}+\frac{1}{36}b^{4}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

Multiply \frac{1}{36} and -18 to get -\frac{1}{2}.

$$16b^{4}-\frac{1}{2}b^{4}b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times 3^{2}\left(b^{4}\right)^{2}$$

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

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times 3^{2}b^{8}$$

Calculate 3 to the power of 2 and get 9.

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times 9b^{8}$$

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

$$16b^{4}-\frac{1}{2}b^{8}+\frac{9}{2}b^{8}$$

Combine -\frac{1}{2}b^{8} and \frac{9}{2}b^{8} to get 4b^{8}.

$$16b^{4}+4b^{8}$$

Expand

$$4b^{8}+16b^{4}$$

Solution Steps

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

$$\left(-2\right)^{4}b^{4}+\left(-\frac{2}{3}b^{2}+\frac{1}{2}b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}+\left(-\frac{2}{3}b^{2}+\frac{1}{2}b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

Combine -\frac{2}{3}b^{2} and \frac{1}{2}b^{2} to get -\frac{1}{6}b^{2}.

$$16b^{4}+\left(-\frac{1}{6}b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}+\left(-\frac{1}{6}\right)^{2}\left(b^{2}\right)^{2}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}+\left(-\frac{1}{6}\right)^{2}b^{4}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

Calculate -\frac{1}{6} to the power of 2 and get \frac{1}{36}.

$$16b^{4}+\frac{1}{36}b^{4}\left(-18\right)b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

Multiply \frac{1}{36} and -18 to get -\frac{1}{2}.

$$16b^{4}-\frac{1}{2}b^{4}b^{4}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times \left(3b^{4}\right)^{2}$$

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

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times 3^{2}\left(b^{4}\right)^{2}$$

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

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times 3^{2}b^{8}$$

Calculate 3 to the power of 2 and get 9.

$$16b^{4}-\frac{1}{2}b^{8}+\frac{1}{2}\times 9b^{8}$$

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

$$16b^{4}-\frac{1}{2}b^{8}+\frac{9}{2}b^{8}$$

Combine -\frac{1}{2}b^{8} and \frac{9}{2}b^{8} to get 4b^{8}.

$$16b^{4}+4b^{8}$$