Solve frac{(3xy^{2})^{3}*(2x^{2}y)^3}{(6xy)^4*(xy^3)^5}

$$\frac { ( 3 x y ^ { 2 } ) ^ { 3 } \times ( 2 x ^ { 2 } y ) ^ { 3 } } { ( 6 x y ) ^ { 4 } \times ( x y ^ { 3 } ) ^ { 5 } }$$

Evaluate

$$\frac{1}{6y^{10}}$$

Solution Steps

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

$$\frac{3^{3}x^{3}\left(y^{2}\right)^{3}\times \left(2x^{2}y\right)^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

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

$$\frac{3^{3}x^{3}y^{6}\times \left(2x^{2}y\right)^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

Calculate 3 to the power of 3 and get 27.

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

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

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

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

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

Calculate 2 to the power of 3 and get 8.

$$\frac{27x^{3}y^{6}\times 8x^{6}y^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

Multiply 27 and 8 to get 216.

$$\frac{216x^{3}y^{6}x^{6}y^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{216x^{9}y^{6}y^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

To multiply powers of the same base, add their exponents. Add 6 and 3 to get 9.

$$\frac{216x^{9}y^{9}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

Expand \left(6xy\right)^{4}.

$$\frac{216x^{9}y^{9}}{6^{4}x^{4}y^{4}\left(xy^{3}\right)^{5}}$$

Calculate 6 to the power of 4 and get 1296.

$$\frac{216x^{9}y^{9}}{1296x^{4}y^{4}\left(xy^{3}\right)^{5}}$$

Expand \left(xy^{3}\right)^{5}.

$$\frac{216x^{9}y^{9}}{1296x^{4}y^{4}x^{5}\left(y^{3}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\frac{216x^{9}y^{9}}{1296x^{4}y^{4}x^{5}y^{15}}$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{216x^{9}y^{9}}{1296x^{9}y^{4}y^{15}}$$

To multiply powers of the same base, add their exponents. Add 4 and 15 to get 19.

$$\frac{216x^{9}y^{9}}{1296x^{9}y^{19}}$$

Cancel out 216x^{9}y^{9} in both numerator and denominator.

$$\frac{1}{6y^{10}}$$

Expand

$$\frac{1}{6y^{10}}$$

Solution Steps

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

$$\frac{3^{3}x^{3}\left(y^{2}\right)^{3}\times \left(2x^{2}y\right)^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

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

$$\frac{3^{3}x^{3}y^{6}\times \left(2x^{2}y\right)^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

Calculate 3 to the power of 3 and get 27.

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

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

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

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

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

Calculate 2 to the power of 3 and get 8.

$$\frac{27x^{3}y^{6}\times 8x^{6}y^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

Multiply 27 and 8 to get 216.

$$\frac{216x^{3}y^{6}x^{6}y^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{216x^{9}y^{6}y^{3}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

To multiply powers of the same base, add their exponents. Add 6 and 3 to get 9.

$$\frac{216x^{9}y^{9}}{\left(6xy\right)^{4}\left(xy^{3}\right)^{5}}$$

Expand \left(6xy\right)^{4}.

$$\frac{216x^{9}y^{9}}{6^{4}x^{4}y^{4}\left(xy^{3}\right)^{5}}$$

Calculate 6 to the power of 4 and get 1296.

$$\frac{216x^{9}y^{9}}{1296x^{4}y^{4}\left(xy^{3}\right)^{5}}$$

Expand \left(xy^{3}\right)^{5}.

$$\frac{216x^{9}y^{9}}{1296x^{4}y^{4}x^{5}\left(y^{3}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\frac{216x^{9}y^{9}}{1296x^{4}y^{4}x^{5}y^{15}}$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{216x^{9}y^{9}}{1296x^{9}y^{4}y^{15}}$$

To multiply powers of the same base, add their exponents. Add 4 and 15 to get 19.

$$\frac{216x^{9}y^{9}}{1296x^{9}y^{19}}$$

Cancel out 216x^{9}y^{9} in both numerator and denominator.

$$\frac{1}{6y^{10}}$$