Solve (-4m^{3}n^5)^2*(-1/2m^2n)^4

$$( - 4 m ^ { 3 } n ^ { 5 } ) ^ { 2 } \cdot ( \frac { - 1 } { 2 } m ^ { 2 } n ) ^ { 4 }$$

Evaluate

$$\left(mn\right)^{14}$$

Solution Steps

Expand \left(-4m^{3}n^{5}\right)^{2}.

$$\left(-4\right)^{2}\left(m^{3}\right)^{2}\left(n^{5}\right)^{2}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

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

$$\left(-4\right)^{2}m^{6}\left(n^{5}\right)^{2}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\left(-4\right)^{2}m^{6}n^{10}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

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

$$16m^{6}n^{10}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$16m^{6}n^{10}\left(-\frac{1}{2}m^{2}n\right)^{4}$$

Expand \left(-\frac{1}{2}m^{2}n\right)^{4}.

$$16m^{6}n^{10}\left(-\frac{1}{2}\right)^{4}\left(m^{2}\right)^{4}n^{4}$$

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

$$16m^{6}n^{10}\left(-\frac{1}{2}\right)^{4}m^{8}n^{4}$$

Calculate -\frac{1}{2} to the power of 4 and get \frac{1}{16}.

$$16m^{6}n^{10}\times \frac{1}{16}m^{8}n^{4}$$

Multiply 16 and \frac{1}{16} to get 1.

$$m^{6}n^{10}m^{8}n^{4}$$

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

$$m^{14}n^{10}n^{4}$$

To multiply powers of the same base, add their exponents. Add 10 and 4 to get 14.

$$m^{14}n^{14}$$

Expand

$$\left(mn\right)^{14}$$

Solution Steps

Expand \left(-4m^{3}n^{5}\right)^{2}.

$$\left(-4\right)^{2}\left(m^{3}\right)^{2}\left(n^{5}\right)^{2}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

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

$$\left(-4\right)^{2}m^{6}\left(n^{5}\right)^{2}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\left(-4\right)^{2}m^{6}n^{10}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

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

$$16m^{6}n^{10}\times \left(\frac{-1}{2}m^{2}n\right)^{4}$$

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$16m^{6}n^{10}\left(-\frac{1}{2}m^{2}n\right)^{4}$$

Expand \left(-\frac{1}{2}m^{2}n\right)^{4}.

$$16m^{6}n^{10}\left(-\frac{1}{2}\right)^{4}\left(m^{2}\right)^{4}n^{4}$$

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

$$16m^{6}n^{10}\left(-\frac{1}{2}\right)^{4}m^{8}n^{4}$$

Calculate -\frac{1}{2} to the power of 4 and get \frac{1}{16}.

$$16m^{6}n^{10}\times \frac{1}{16}m^{8}n^{4}$$

Multiply 16 and \frac{1}{16} to get 1.

$$m^{6}n^{10}m^{8}n^{4}$$

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

$$m^{14}n^{10}n^{4}$$

To multiply powers of the same base, add their exponents. Add 10 and 4 to get 14.

$$m^{14}n^{14}$$