Solve {left(frac{8{x}^-frac{2{3}}{b}^-frac{1{2}}}{27x{b}^frac{3{4}}}right)}^frac{2{3}}

$${ \left( \frac{ 8 { x }^{ - \frac{ 2 }{ 3 } } { b }^{ - \frac{ 1 }{ 2 } } }{ 27x { b }^{ \frac{ 3 }{ 4 } } } \right) }^{ \frac{ 2 }{ 3 } }$$

Evaluate

$$\frac{4}{9b^{\frac{5}{6}}x^{\frac{10}{9}}}$$

Solution Steps

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

$$\left(\frac{8}{27b^{\frac{5}{4}}x^{\frac{5}{3}}}\right)^{\frac{2}{3}}$$

To raise \frac{8}{27b^{\frac{5}{4}}x^{\frac{5}{3}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8^{\frac{2}{3}}}{\left(27b^{\frac{5}{4}}x^{\frac{5}{3}}\right)^{\frac{2}{3}}}$$

Calculate 8 to the power of \frac{2}{3} and get 4.

$$\frac{4}{\left(27b^{\frac{5}{4}}x^{\frac{5}{3}}\right)^{\frac{2}{3}}}$$

Expand \left(27b^{\frac{5}{4}}x^{\frac{5}{3}}\right)^{\frac{2}{3}}.

$$\frac{4}{27^{\frac{2}{3}}\left(b^{\frac{5}{4}}\right)^{\frac{2}{3}}\left(x^{\frac{5}{3}}\right)^{\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{4} and \frac{2}{3} to get \frac{5}{6}.

$$\frac{4}{27^{\frac{2}{3}}b^{\frac{5}{6}}\left(x^{\frac{5}{3}}\right)^{\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{3} and \frac{2}{3} to get \frac{10}{9}.

$$\frac{4}{27^{\frac{2}{3}}b^{\frac{5}{6}}x^{\frac{10}{9}}}$$

Calculate 27 to the power of \frac{2}{3} and get 9.

$$\frac{4}{9b^{\frac{5}{6}}x^{\frac{10}{9}}}$$

Differentiate w.r.t. x

$$-\frac{40}{81b^{\frac{5}{6}}x^{\frac{19}{9}}}$$