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

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

Evaluate

$$\frac{4n^{37}}{m^{25}}$$

Solution Steps

Expand \left(2m^{-3}n^{6}\right)^{6}.

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

To raise a power to another power, multiply the exponents. Multiply -3 and 6 to get -18.

$$\frac{2^{6}m^{-18}\left(n^{6}\right)^{6}}{\left(2m^{-2}n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 6 to get 36.

$$\frac{2^{6}m^{-18}n^{36}}{\left(2m^{-2}n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{64m^{-18}n^{36}}{\left(2m^{-2}n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{2^{-1}\left(m^{-2}\right)^{-1}\left(n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{2^{-1}m^{2}\left(n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{2^{-1}m^{2}n^{-6}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{\frac{1}{2}m^{2}n^{-6}\times \left(2mn\right)^{5}}$$

Expand \left(2mn\right)^{5}.

$$\frac{64m^{-18}n^{36}}{\frac{1}{2}m^{2}n^{-6}\times 2^{5}m^{5}n^{5}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{64m^{-18}n^{36}}{\frac{1}{2}m^{2}n^{-6}\times 32m^{5}n^{5}}$$

Multiply \frac{1}{2} and 32 to get 16.

$$\frac{64m^{-18}n^{36}}{16m^{2}n^{-6}m^{5}n^{5}}$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{64m^{-18}n^{36}}{16m^{7}n^{-6}n^{5}}$$

To multiply powers of the same base, add their exponents. Add -6 and 5 to get -1.

$$\frac{64m^{-18}n^{36}}{16m^{7}n^{-1}}$$

Cancel out 16 in both numerator and denominator.

$$\frac{4m^{-18}n^{36}}{\frac{1}{n}m^{7}}$$

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

$$\frac{4m^{-18}n^{37}}{m^{7}}$$

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

$$\frac{4n^{37}}{m^{25}}$$

Expand

$$\frac{4n^{37}}{m^{25}}$$

Solution Steps

Expand \left(2m^{-3}n^{6}\right)^{6}.

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

To raise a power to another power, multiply the exponents. Multiply -3 and 6 to get -18.

$$\frac{2^{6}m^{-18}\left(n^{6}\right)^{6}}{\left(2m^{-2}n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 6 to get 36.

$$\frac{2^{6}m^{-18}n^{36}}{\left(2m^{-2}n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{64m^{-18}n^{36}}{\left(2m^{-2}n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{2^{-1}\left(m^{-2}\right)^{-1}\left(n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{2^{-1}m^{2}\left(n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{2^{-1}m^{2}n^{-6}\times \left(2mn\right)^{5}}$$

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

$$\frac{64m^{-18}n^{36}}{\frac{1}{2}m^{2}n^{-6}\times \left(2mn\right)^{5}}$$

Expand \left(2mn\right)^{5}.

$$\frac{64m^{-18}n^{36}}{\frac{1}{2}m^{2}n^{-6}\times 2^{5}m^{5}n^{5}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{64m^{-18}n^{36}}{\frac{1}{2}m^{2}n^{-6}\times 32m^{5}n^{5}}$$

Multiply \frac{1}{2} and 32 to get 16.

$$\frac{64m^{-18}n^{36}}{16m^{2}n^{-6}m^{5}n^{5}}$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{64m^{-18}n^{36}}{16m^{7}n^{-6}n^{5}}$$

To multiply powers of the same base, add their exponents. Add -6 and 5 to get -1.

$$\frac{64m^{-18}n^{36}}{16m^{7}n^{-1}}$$

Cancel out 16 in both numerator and denominator.

$$\frac{4m^{-18}n^{36}}{\frac{1}{n}m^{7}}$$

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

$$\frac{4m^{-18}n^{37}}{m^{7}}$$

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

$$\frac{4n^{37}}{m^{25}}$$