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

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

Evaluate

$$mn^{5}$$

Solution Steps

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

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and 4 to get 2.

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

To raise a power to another power, multiply the exponents. Multiply \frac{1}{4} and 4 to get 1.

$$\frac{m^{3}n^{6}}{m^{2}n^{1}}$$

Calculate n to the power of 1 and get n.

$$\frac{m^{3}n^{6}}{m^{2}n}$$

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

$$mn^{5}$$

Expand

$$mn^{5}$$

Solution Steps

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

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and 4 to get 2.

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

To raise a power to another power, multiply the exponents. Multiply \frac{1}{4} and 4 to get 1.

$$\frac{m^{3}n^{6}}{m^{2}n^{1}}$$

Calculate n to the power of 1 and get n.

$$\frac{m^{3}n^{6}}{m^{2}n}$$

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

$$mn^{5}$$