Solve (8a^3/27b^6)^frac{1{3}}

$$( \frac { 8 a ^ { 3 } } { 27 b ^ { 6 } } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$\frac{2a}{3b^{2}}$$

Solution Steps

To raise \frac{8a^{3}}{27b^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8a^{3}\right)^{\frac{1}{3}}}{\left(27b^{6}\right)^{\frac{1}{3}}}$$

Expand \left(8a^{3}\right)^{\frac{1}{3}}.

$$\frac{8^{\frac{1}{3}}\left(a^{3}\right)^{\frac{1}{3}}}{\left(27b^{6}\right)^{\frac{1}{3}}}$$

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

$$\frac{8^{\frac{1}{3}}a^{1}}{\left(27b^{6}\right)^{\frac{1}{3}}}$$

Calculate 8 to the power of \frac{1}{3} and get 2.

$$\frac{2a^{1}}{\left(27b^{6}\right)^{\frac{1}{3}}}$$

Calculate a to the power of 1 and get a.

$$\frac{2a}{\left(27b^{6}\right)^{\frac{1}{3}}}$$

Expand \left(27b^{6}\right)^{\frac{1}{3}}.

$$\frac{2a}{27^{\frac{1}{3}}\left(b^{6}\right)^{\frac{1}{3}}}$$

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

$$\frac{2a}{27^{\frac{1}{3}}b^{2}}$$

Calculate 27 to the power of \frac{1}{3} and get 3.

$$\frac{2a}{3b^{2}}$$

Differentiate w.r.t. a

$$\frac{2}{3b^{2}}$$