Solve frac{(3x)^{3}y^{2}}{(6x^{2}y^{2})^{2}}div(5xy)^2/(27x^9y^9)^frac{1{3}}

$$\frac { ( 3 x ) ^ { 3 } y ^ { 2 } } { ( 6 x ^ { 2 } y ^ { 2 } ) ^ { 2 } } \div \frac { ( 5 x y ) ^ { 2 } } { ( 27 x ^ { 9 } y ^ { 9 } ) ^ { \frac { 1 } { 3 } } }$$

Evaluate

$$\frac{9}{100y}$$

Solution Steps

Divide \frac{\left(3x\right)^{3}y^{2}}{\left(6x^{2}y^{2}\right)^{2}} by \frac{\left(5xy\right)^{2}}{\left(27x^{9}y^{9}\right)^{\frac{1}{3}}} by multiplying \frac{\left(3x\right)^{3}y^{2}}{\left(6x^{2}y^{2}\right)^{2}} by the reciprocal of \frac{\left(5xy\right)^{2}}{\left(27x^{9}y^{9}\right)^{\frac{1}{3}}}.

$$\frac{\left(3x\right)^{3}y^{2}\times \left(27x^{9}y^{9}\right)^{\frac{1}{3}}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

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

$$\frac{3^{3}x^{3}y^{2}\times \left(27x^{9}y^{9}\right)^{\frac{1}{3}}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{27x^{3}y^{2}\times \left(27x^{9}y^{9}\right)^{\frac{1}{3}}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

Expand \left(27x^{9}y^{9}\right)^{\frac{1}{3}}.

$$\frac{27x^{3}y^{2}\times 27^{\frac{1}{3}}\left(x^{9}\right)^{\frac{1}{3}}\left(y^{9}\right)^{\frac{1}{3}}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 9 and \frac{1}{3} to get 3.

$$\frac{27x^{3}y^{2}\times 27^{\frac{1}{3}}x^{3}\left(y^{9}\right)^{\frac{1}{3}}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 9 and \frac{1}{3} to get 3.

$$\frac{27x^{3}y^{2}\times 27^{\frac{1}{3}}x^{3}y^{3}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

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

$$\frac{27x^{3}y^{2}\times 3x^{3}y^{3}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

Multiply 27 and 3 to get 81.

$$\frac{81x^{3}y^{2}x^{3}y^{3}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{81x^{6}y^{2}y^{3}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{81x^{6}y^{5}}{\left(6x^{2}y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

Expand \left(6x^{2}y^{2}\right)^{2}.

$$\frac{81x^{6}y^{5}}{6^{2}\left(x^{2}\right)^{2}\left(y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{81x^{6}y^{5}}{6^{2}x^{4}\left(y^{2}\right)^{2}\times \left(5xy\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{81x^{6}y^{5}}{6^{2}x^{4}y^{4}\times \left(5xy\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{81x^{6}y^{5}}{36x^{4}y^{4}\times \left(5xy\right)^{2}}$$

Expand \left(5xy\right)^{2}.

$$\frac{81x^{6}y^{5}}{36x^{4}y^{4}\times 5^{2}x^{2}y^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{81x^{6}y^{5}}{36x^{4}y^{4}\times 25x^{2}y^{2}}$$

Multiply 36 and 25 to get 900.

$$\frac{81x^{6}y^{5}}{900x^{4}y^{4}x^{2}y^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{81x^{6}y^{5}}{900x^{6}y^{4}y^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{81x^{6}y^{5}}{900x^{6}y^{6}}$$

Cancel out 9y^{5}x^{6} in both numerator and denominator.

$$\frac{9}{100y}$$

Differentiate w.r.t. y

$$-\frac{9}{100y^{2}}$$