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

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

Evaluate

$$64b^{6}a^{14}$$

Solution Steps

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

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

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

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

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

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

Calculate 4 to the power of 4 and get 256.

$$256a^{8}b^{8}\times \left(\frac{-a^{3}}{2b}\right)^{2}$$

To raise \frac{-a^{3}}{2b} to a power, raise both numerator and denominator to the power and then divide.

$$256a^{8}b^{8}\times \frac{\left(-a^{3}\right)^{2}}{\left(2b\right)^{2}}$$

Express 256\times \frac{\left(-a^{3}\right)^{2}}{\left(2b\right)^{2}} as a single fraction.

$$\frac{256\left(-a^{3}\right)^{2}}{\left(2b\right)^{2}}a^{8}b^{8}$$

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

$$\frac{256\left(a^{3}\right)^{2}}{\left(2b\right)^{2}}a^{8}b^{8}$$

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

$$\frac{256\left(a^{3}\right)^{2}}{2^{2}b^{2}}a^{8}b^{8}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{256\left(a^{3}\right)^{2}}{4b^{2}}a^{8}b^{8}$$

Cancel out 4 in both numerator and denominator.

$$\frac{64\left(a^{3}\right)^{2}}{b^{2}}a^{8}b^{8}$$

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

$$\frac{64a^{6}}{b^{2}}a^{8}b^{8}$$

Express \frac{64a^{6}}{b^{2}}a^{8} as a single fraction.

$$\frac{64a^{6}a^{8}}{b^{2}}b^{8}$$

Express \frac{64a^{6}a^{8}}{b^{2}}b^{8} as a single fraction.

$$\frac{64a^{6}a^{8}b^{8}}{b^{2}}$$

Cancel out b^{2} in both numerator and denominator.

$$64a^{6}b^{6}a^{8}$$

To multiply powers of the same base, add their exponents. Add 6 and 8 to get 14.

$$64a^{14}b^{6}$$

Expand

$$64b^{6}a^{14}$$

Solution Steps

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

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

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

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

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

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

Calculate 4 to the power of 4 and get 256.

$$256a^{8}b^{8}\times \left(\frac{-a^{3}}{2b}\right)^{2}$$

To raise \frac{-a^{3}}{2b} to a power, raise both numerator and denominator to the power and then divide.

$$256a^{8}b^{8}\times \frac{\left(-a^{3}\right)^{2}}{\left(2b\right)^{2}}$$

Express 256\times \frac{\left(-a^{3}\right)^{2}}{\left(2b\right)^{2}} as a single fraction.

$$\frac{256\left(-a^{3}\right)^{2}}{\left(2b\right)^{2}}a^{8}b^{8}$$

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

$$\frac{256\left(a^{3}\right)^{2}}{\left(2b\right)^{2}}a^{8}b^{8}$$

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

$$\frac{256\left(a^{3}\right)^{2}}{2^{2}b^{2}}a^{8}b^{8}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{256\left(a^{3}\right)^{2}}{4b^{2}}a^{8}b^{8}$$

Cancel out 4 in both numerator and denominator.

$$\frac{64\left(a^{3}\right)^{2}}{b^{2}}a^{8}b^{8}$$

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

$$\frac{64a^{6}}{b^{2}}a^{8}b^{8}$$

Express \frac{64a^{6}}{b^{2}}a^{8} as a single fraction.

$$\frac{64a^{6}a^{8}}{b^{2}}b^{8}$$

Express \frac{64a^{6}a^{8}}{b^{2}}b^{8} as a single fraction.

$$\frac{64a^{6}a^{8}b^{8}}{b^{2}}$$

Cancel out b^{2} in both numerator and denominator.

$$64a^{6}b^{6}a^{8}$$

To multiply powers of the same base, add their exponents. Add 6 and 8 to get 14.

$$64a^{14}b^{6}$$