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

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

Evaluate

$$32a^{6}b^{10}$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$\frac{1}{8}a^{6}b^{6}\left(-4b\right)^{4}$$

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

$$\frac{1}{8}a^{6}b^{6}\left(-4\right)^{4}b^{4}$$

Calculate -4 to the power of 4 and get 256.

$$\frac{1}{8}a^{6}b^{6}\times 256b^{4}$$

Multiply \frac{1}{8} and 256 to get 32.

$$32a^{6}b^{6}b^{4}$$

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$32a^{6}b^{10}$$

Expand

$$32a^{6}b^{10}$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$\frac{1}{8}a^{6}b^{6}\left(-4b\right)^{4}$$

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

$$\frac{1}{8}a^{6}b^{6}\left(-4\right)^{4}b^{4}$$

Calculate -4 to the power of 4 and get 256.

$$\frac{1}{8}a^{6}b^{6}\times 256b^{4}$$

Multiply \frac{1}{8} and 256 to get 32.

$$32a^{6}b^{6}b^{4}$$

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$32a^{6}b^{10}$$