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

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

Evaluate

$$\frac{mna^{\frac{13}{2}}}{b^{2}}$$

Solution Steps

Calculate n to the power of 1 and get n.

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

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

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

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

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

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

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

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

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

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

$$\frac{ma^{\frac{1}{2}}b^{-1}}{9nb\times 3^{-2}n^{-2}a^{-6}}$$

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

$$\frac{ma^{\frac{1}{2}}b^{-1}}{9nb\times \frac{1}{9}n^{-2}a^{-6}}$$

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

$$\frac{ma^{\frac{1}{2}}b^{-1}}{nbn^{-2}a^{-6}}$$

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

$$\frac{ma^{\frac{1}{2}}b^{-1}}{n^{-1}ba^{-6}}$$

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

$$\frac{\frac{1}{b}ma^{\frac{13}{2}}}{\frac{1}{n}b}$$

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

$$\frac{ma^{\frac{13}{2}}}{\frac{1}{n}b^{2}}$$

Express \frac{1}{n}b^{2} as a single fraction.

$$\frac{ma^{\frac{13}{2}}}{\frac{b^{2}}{n}}$$

Divide ma^{\frac{13}{2}} by \frac{b^{2}}{n} by multiplying ma^{\frac{13}{2}} by the reciprocal of \frac{b^{2}}{n}.

$$\frac{ma^{\frac{13}{2}}n}{b^{2}}$$

Differentiate w.r.t. m

$$\frac{na^{\frac{13}{2}}}{b^{2}}$$