Solve frac{(2m^{1/3}*n^{5/6})^{6}}{(2m^-2*n^6)^-1}(2mn)^6

$$\frac { ( 2 m ^ { 1 / 3 } \cdot n ^ { 5 / 6 } ) ^ { 6 } } { ( 2 m ^ { - 2 } \cdot n ^ { 6 } ) ^ { - 1 } } ( 2 m n ) ^ { 6 }$$

Evaluate

$$8192m^{6}n^{17}$$

Solution Steps

Expand \left(2m^{\frac{1}{3}}n^{\frac{5}{6}}\right)^{6}.

$$\frac{2^{6}\left(m^{\frac{1}{3}}\right)^{6}\left(n^{\frac{5}{6}}\right)^{6}}{\left(2m^{-2}n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and 6 to get 2.

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

To raise a power to another power, multiply the exponents. Multiply \frac{5}{6} and 6 to get 5.

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

Calculate 2 to the power of 6 and get 64.

$$\frac{64m^{2}n^{5}}{\left(2m^{-2}n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

Expand \left(2m^{-2}n^{6}\right)^{-1}.

$$\frac{64m^{2}n^{5}}{2^{-1}\left(m^{-2}\right)^{-1}\left(n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$\frac{64m^{2}n^{5}}{2^{-1}m^{2}\left(n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -1 to get -6.

$$\frac{64m^{2}n^{5}}{2^{-1}m^{2}n^{-6}}\times \left(2mn\right)^{6}$$

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

$$\frac{64m^{2}n^{5}}{\frac{1}{2}m^{2}n^{-6}}\times \left(2mn\right)^{6}$$

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

$$\frac{64n^{5}}{\frac{1}{2}n^{-6}}\times \left(2mn\right)^{6}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{64n^{11}}{\frac{1}{2}}\times \left(2mn\right)^{6}$$

Divide 64n^{11} by \frac{1}{2} by multiplying 64n^{11} by the reciprocal of \frac{1}{2}.

$$64n^{11}\times 2\times \left(2mn\right)^{6}$$

Multiply 64 and 2 to get 128.

$$128n^{11}\times \left(2mn\right)^{6}$$

Expand \left(2mn\right)^{6}.

$$128n^{11}\times 2^{6}m^{6}n^{6}$$

Calculate 2 to the power of 6 and get 64.

$$128n^{11}\times 64m^{6}n^{6}$$

Multiply 128 and 64 to get 8192.

$$8192n^{11}m^{6}n^{6}$$

To multiply powers of the same base, add their exponents. Add 11 and 6 to get 17.

$$8192n^{17}m^{6}$$

Expand

$$8192m^{6}n^{17}$$

Solution Steps

Expand \left(2m^{\frac{1}{3}}n^{\frac{5}{6}}\right)^{6}.

$$\frac{2^{6}\left(m^{\frac{1}{3}}\right)^{6}\left(n^{\frac{5}{6}}\right)^{6}}{\left(2m^{-2}n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{3} and 6 to get 2.

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

To raise a power to another power, multiply the exponents. Multiply \frac{5}{6} and 6 to get 5.

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

Calculate 2 to the power of 6 and get 64.

$$\frac{64m^{2}n^{5}}{\left(2m^{-2}n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

Expand \left(2m^{-2}n^{6}\right)^{-1}.

$$\frac{64m^{2}n^{5}}{2^{-1}\left(m^{-2}\right)^{-1}\left(n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$\frac{64m^{2}n^{5}}{2^{-1}m^{2}\left(n^{6}\right)^{-1}}\times \left(2mn\right)^{6}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -1 to get -6.

$$\frac{64m^{2}n^{5}}{2^{-1}m^{2}n^{-6}}\times \left(2mn\right)^{6}$$

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

$$\frac{64m^{2}n^{5}}{\frac{1}{2}m^{2}n^{-6}}\times \left(2mn\right)^{6}$$

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

$$\frac{64n^{5}}{\frac{1}{2}n^{-6}}\times \left(2mn\right)^{6}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{64n^{11}}{\frac{1}{2}}\times \left(2mn\right)^{6}$$

Divide 64n^{11} by \frac{1}{2} by multiplying 64n^{11} by the reciprocal of \frac{1}{2}.

$$64n^{11}\times 2\times \left(2mn\right)^{6}$$

Multiply 64 and 2 to get 128.

$$128n^{11}\times \left(2mn\right)^{6}$$

Expand \left(2mn\right)^{6}.

$$128n^{11}\times 2^{6}m^{6}n^{6}$$

Calculate 2 to the power of 6 and get 64.

$$128n^{11}\times 64m^{6}n^{6}$$

Multiply 128 and 64 to get 8192.

$$8192n^{11}m^{6}n^{6}$$

To multiply powers of the same base, add their exponents. Add 11 and 6 to get 17.

$$8192n^{17}m^{6}$$