Solve {left(frac{1/6xy^3right)}^2{left(-2/3x^2yright)}^2}{{left(-1/3x^2y^2right)}^3}

$$\frac{ { \left( \frac{ 1 }{ 6 } x { y }^{ 3 } \right) }^{ 2 } { \left(- \frac{ 2 }{ 3 } { x }^{ 2 } y \right) }^{ 2 } }{ { \left(- \frac{ 1 }{ 3 } { x }^{ 2 } { y }^{ 2 } \right) }^{ 3 } }$$

Evaluate

$$-\frac{y^{2}}{3}$$

Solution Steps

Expand \left(\frac{1}{6}xy^{3}\right)^{2}.

$$\frac{\left(\frac{1}{6}\right)^{2}x^{2}\left(y^{3}\right)^{2}\left(-\frac{2}{3}x^{2}y\right)^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\left(\frac{1}{6}\right)^{2}x^{2}y^{6}\left(-\frac{2}{3}x^{2}y\right)^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

Calculate \frac{1}{6} to the power of 2 and get \frac{1}{36}.

$$\frac{\frac{1}{36}x^{2}y^{6}\left(-\frac{2}{3}x^{2}y\right)^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{36}x^{2}y^{6}\left(-\frac{2}{3}\right)^{2}\left(x^{2}\right)^{2}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{36}x^{2}y^{6}\left(-\frac{2}{3}\right)^{2}x^{4}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

Calculate -\frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{1}{36}x^{2}y^{6}\times \frac{4}{9}x^{4}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

Multiply \frac{1}{36} and \frac{4}{9} to get \frac{1}{81}.

$$\frac{\frac{1}{81}x^{2}y^{6}x^{4}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{6}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}\right)^{3}\left(x^{2}\right)^{3}\left(y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}\right)^{3}x^{6}\left(y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}\right)^{3}x^{6}y^{6}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{-\frac{1}{27}x^{6}y^{6}}$$

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

$$\frac{\frac{1}{81}y^{2}}{-\frac{1}{27}}$$

Divide \frac{1}{81}y^{2} by -\frac{1}{27} by multiplying \frac{1}{81}y^{2} by the reciprocal of -\frac{1}{27}.

$$\frac{\frac{1}{81}y^{2}\times 27}{-1}$$

Multiply \frac{1}{81} and 27 to get \frac{1}{3}.

$$\frac{\frac{1}{3}y^{2}}{-1}$$

Anything divided by -1 gives its opposite.

$$-\frac{1}{3}y^{2}$$

Expand

$$-\frac{y^{2}}{3}$$

Solution Steps

Expand \left(\frac{1}{6}xy^{3}\right)^{2}.

$$\frac{\left(\frac{1}{6}\right)^{2}x^{2}\left(y^{3}\right)^{2}\left(-\frac{2}{3}x^{2}y\right)^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\left(\frac{1}{6}\right)^{2}x^{2}y^{6}\left(-\frac{2}{3}x^{2}y\right)^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

Calculate \frac{1}{6} to the power of 2 and get \frac{1}{36}.

$$\frac{\frac{1}{36}x^{2}y^{6}\left(-\frac{2}{3}x^{2}y\right)^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{36}x^{2}y^{6}\left(-\frac{2}{3}\right)^{2}\left(x^{2}\right)^{2}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{36}x^{2}y^{6}\left(-\frac{2}{3}\right)^{2}x^{4}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

Calculate -\frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{1}{36}x^{2}y^{6}\times \frac{4}{9}x^{4}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

Multiply \frac{1}{36} and \frac{4}{9} to get \frac{1}{81}.

$$\frac{\frac{1}{81}x^{2}y^{6}x^{4}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{6}y^{2}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}x^{2}y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}\right)^{3}\left(x^{2}\right)^{3}\left(y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}\right)^{3}x^{6}\left(y^{2}\right)^{3}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{\left(-\frac{1}{3}\right)^{3}x^{6}y^{6}}$$

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

$$\frac{\frac{1}{81}x^{6}y^{8}}{-\frac{1}{27}x^{6}y^{6}}$$

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

$$\frac{\frac{1}{81}y^{2}}{-\frac{1}{27}}$$

Divide \frac{1}{81}y^{2} by -\frac{1}{27} by multiplying \frac{1}{81}y^{2} by the reciprocal of -\frac{1}{27}.

$$\frac{\frac{1}{81}y^{2}\times 27}{-1}$$

Multiply \frac{1}{81} and 27 to get \frac{1}{3}.

$$\frac{\frac{1}{3}y^{2}}{-1}$$

Anything divided by -1 gives its opposite.

$$-\frac{1}{3}y^{2}$$