Solve frac{(x^{-3}y)^{3}}{(3y^{2/3})^{3}}divfrac{18(x^{3}y^{-2})^frac{1{2}}}{(2x^frac{3{4}}y^frac{1{2}})^2}

$$\frac { ( x ^ { - 3 } y ) ^ { 3 } } { ( 3 y ^ { \frac { 2 } { 3 } } ) ^ { 3 } } \div \frac { 18 ( x ^ { 3 } y ^ { - 2 } ) ^ { \frac { 1 } { 2 } } } { ( 2 x ^ { \frac { 3 } { 4 } } y ^ { \frac { 1 } { 2 } } ) ^ { 2 } }$$

Evaluate

$$\frac{2y^{3}}{243x^{9}}$$

Solution Steps

Divide \frac{\left(x^{-3}y\right)^{3}}{\left(3y^{\frac{2}{3}}\right)^{3}} by \frac{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}} by multiplying \frac{\left(x^{-3}y\right)^{3}}{\left(3y^{\frac{2}{3}}\right)^{3}} by the reciprocal of \frac{18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}{\left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}.

$$\frac{\left(x^{-3}y\right)^{3}\times \left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Expand \left(x^{-3}y\right)^{3}.

$$\frac{\left(x^{-3}\right)^{3}y^{3}\times \left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{x^{-9}y^{3}\times \left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Expand \left(2x^{\frac{3}{4}}y^{\frac{1}{2}}\right)^{2}.

$$\frac{x^{-9}y^{3}\times 2^{2}\left(x^{\frac{3}{4}}\right)^{2}\left(y^{\frac{1}{2}}\right)^{2}}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{x^{-9}y^{3}\times 2^{2}x^{\frac{3}{2}}\left(y^{\frac{1}{2}}\right)^{2}}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{x^{-9}y^{3}\times 2^{2}x^{\frac{3}{2}}y^{1}}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{x^{-9}y^{3}\times 4x^{\frac{3}{2}}y^{1}}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Calculate y to the power of 1 and get y.

$$\frac{x^{-9}y^{3}\times 4x^{\frac{3}{2}}y}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

To multiply powers of the same base, add their exponents. Add -9 and \frac{3}{2} to get -\frac{15}{2}.

$$\frac{x^{-\frac{15}{2}}y^{3}\times 4y}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{\left(3y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{3^{3}\left(y^{\frac{2}{3}}\right)^{3}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{3^{3}y^{2}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{27y^{2}\times 18\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Multiply 27 and 18 to get 486.

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{486y^{2}\left(x^{3}y^{-2}\right)^{\frac{1}{2}}}$$

Expand \left(x^{3}y^{-2}\right)^{\frac{1}{2}}.

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{486y^{2}\left(x^{3}\right)^{\frac{1}{2}}\left(y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{486y^{2}x^{\frac{3}{2}}\left(y^{-2}\right)^{\frac{1}{2}}}$$

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

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{486y^{2}x^{\frac{3}{2}}y^{-1}}$$

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

$$\frac{x^{-\frac{15}{2}}y^{4}\times 4}{486y^{1}x^{\frac{3}{2}}}$$

Cancel out 2y in both numerator and denominator.

$$\frac{2x^{-\frac{15}{2}}y^{3}}{243x^{\frac{3}{2}}}$$

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

$$\frac{2y^{3}}{243x^{9}}$$

Differentiate w.r.t. x

$$-\frac{2y^{3}}{27x^{10}}$$