Solve frac{ma^{1/3}}{9nb}div(3n^1/2a^3/2/b^-1/2)^-2

$$\frac { m a ^ { 1 / 3 } } { 9 n b } \div ( \frac { 3 n ^ { 1 / 2 } a ^ { 3 / 2 } } { b ^ { - 1 / 2 } } ) ^ { - 2 }$$

Evaluate

$$ma^{\frac{10}{3}}$$

Solution Steps

To raise \frac{3n^{\frac{1}{2}}a^{\frac{3}{2}}}{b^{-\frac{1}{2}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{ma^{\frac{1}{3}}}{9nb}}{\frac{\left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}{\left(b^{-\frac{1}{2}}\right)^{-2}}}$$

Divide \frac{ma^{\frac{1}{3}}}{9nb} by \frac{\left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}{\left(b^{-\frac{1}{2}}\right)^{-2}} by multiplying \frac{ma^{\frac{1}{3}}}{9nb} by the reciprocal of \frac{\left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}{\left(b^{-\frac{1}{2}}\right)^{-2}}.

$$\frac{ma^{\frac{1}{3}}\left(b^{-\frac{1}{2}}\right)^{-2}}{9nb\times \left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}$$

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

$$\frac{ma^{\frac{1}{3}}b^{1}}{9nb\times \left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}$$

Calculate b to the power of 1 and get b.

$$\frac{ma^{\frac{1}{3}}b}{9nb\times \left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}$$

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

$$\frac{ma^{\frac{1}{3}}b}{9nb\times 3^{-2}\left(n^{\frac{1}{2}}\right)^{-2}\left(a^{\frac{3}{2}}\right)^{-2}}$$

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

$$\frac{ma^{\frac{1}{3}}b}{9nb\times 3^{-2}n^{-1}\left(a^{\frac{3}{2}}\right)^{-2}}$$

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

$$\frac{ma^{\frac{1}{3}}b}{9nb\times 3^{-2}n^{-1}a^{-3}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{ma^{\frac{1}{3}}b}{9nb\times \frac{1}{9}n^{-1}a^{-3}}$$

Multiply 9 and \frac{1}{9} to get 1.

$$\frac{ma^{\frac{1}{3}}b}{nbn^{-1}a^{-3}}$$

Multiply n and n^{-1} to get 1.

$$\frac{ma^{\frac{1}{3}}b}{ba^{-3}}$$

Cancel out b in both numerator and denominator.

$$\frac{\sqrt[3]{a}m}{a^{-3}}$$

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

$$ma^{\frac{10}{3}}$$

Differentiate w.r.t. m

$$a^{\frac{10}{3}}$$