Solve frac{(m^{-1}n^{3})^2*4mn^3}{(2mn^2)^3}

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

Evaluate

$$\frac{n^{3}}{2m^{4}}$$

Solution Steps

Expand \left(m^{-1}n^{3}\right)^{2}.

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

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

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

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

$$\frac{m^{-2}n^{6}\times 4mn^{3}}{\left(2mn^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$\frac{m^{-1}n^{6}\times 4n^{3}}{\left(2mn^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 3 to get 9.

$$\frac{m^{-1}n^{9}\times 4}{\left(2mn^{2}\right)^{3}}$$

Expand \left(2mn^{2}\right)^{3}.

$$\frac{m^{-1}n^{9}\times 4}{2^{3}m^{3}\left(n^{2}\right)^{3}}$$

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

$$\frac{m^{-1}n^{9}\times 4}{2^{3}m^{3}n^{6}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{m^{-1}n^{9}\times 4}{8m^{3}n^{6}}$$

Cancel out 4n^{6} in both numerator and denominator.

$$\frac{\frac{1}{m}n^{3}}{2m^{3}}$$

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

$$\frac{n^{3}}{2m^{4}}$$

Expand

$$\frac{n^{3}}{2m^{4}}$$

Solution Steps

Expand \left(m^{-1}n^{3}\right)^{2}.

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

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

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

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

$$\frac{m^{-2}n^{6}\times 4mn^{3}}{\left(2mn^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$\frac{m^{-1}n^{6}\times 4n^{3}}{\left(2mn^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 3 to get 9.

$$\frac{m^{-1}n^{9}\times 4}{\left(2mn^{2}\right)^{3}}$$

Expand \left(2mn^{2}\right)^{3}.

$$\frac{m^{-1}n^{9}\times 4}{2^{3}m^{3}\left(n^{2}\right)^{3}}$$

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

$$\frac{m^{-1}n^{9}\times 4}{2^{3}m^{3}n^{6}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{m^{-1}n^{9}\times 4}{8m^{3}n^{6}}$$

Cancel out 4n^{6} in both numerator and denominator.

$$\frac{\frac{1}{m}n^{3}}{2m^{3}}$$

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

$$\frac{n^{3}}{2m^{4}}$$