Solve {[frac{1/3xy^{2}-(2/3xy)^{2}:(1/3x)]^{3}-(3/2xy^{2})^{2}*frac{1}{3}xy^{2}}^2}{[7xy^3*(-frac{1}{2}xy^2)^-1]}

$$\frac { \{ [ \frac { 1 } { 3 } x y ^ { 2 } - ( \frac { 2 } { 3 } x y ) ^ { 2 } : ( \frac { 1 } { 3 } x ) ] ^ { 3 } - ( \frac { 3 } { 2 } x y ^ { 2 } ) ^ { 2 } \cdot \frac { 1 } { 3 } x y ^ { 2 } \} ^ { 2 } } { [ 7 x y ^ { 3 } \cdot ( - \frac { 1 } { 2 } x y ^ { 2 } ) ^ { - 1 } ] }$$

Evaluate

$$-\frac{7x^{6}y^{11}}{32}$$

Solution Steps

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{\left(\frac{2}{3}\right)^{2}x^{2}y^{2}}{\frac{1}{3}x}\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{\frac{4}{9}x^{2}y^{2}}{\frac{1}{3}x}\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{\frac{4}{9}xy^{2}}{\frac{1}{3}}\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

Divide \frac{4}{9}xy^{2} by \frac{1}{3} by multiplying \frac{4}{9}xy^{2} by the reciprocal of \frac{1}{3}.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\left(\frac{3}{2}\right)^{2}x^{2}\left(y^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\left(\frac{3}{2}\right)^{2}x^{2}y^{4}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{9}{4}x^{2}y^{4}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

Multiply \frac{9}{4} and \frac{1}{3} to get \frac{3}{4}.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{2}y^{4}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{4}y^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}\right)^{-1}x^{-1}\left(y^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}\right)^{-1}x^{-1}y^{-2}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-2\right)x^{-1}y^{-2}}$$

Multiply 7 and -2 to get -14.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14xy^{3}x^{-1}y^{-2}}$$

Multiply x and x^{-1} to get 1.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y^{3}y^{-2}}$$

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y^{1}}$$

Multiply \frac{4}{9} and 3 to get \frac{4}{3}.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{3}xy^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y^{1}}$$

Calculate y to the power of 1 and get y.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{3}xy^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Combine \frac{1}{3}xy^{2} and -\frac{4}{3}xy^{2} to get -xy^{2}.

$$\frac{\left(\left(-xy^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

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

$$\frac{\left(\left(-1\right)^{3}x^{3}\left(y^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

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

$$\frac{\left(\left(-1\right)^{3}x^{3}y^{6}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Calculate -1 to the power of 3 and get -1.

$$\frac{\left(-x^{3}y^{6}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Combine -x^{3}y^{6} and -\frac{3}{4}x^{3}y^{6} to get -\frac{7}{4}x^{3}y^{6}.

$$\frac{\left(-\frac{7}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Expand \left(-\frac{7}{4}x^{3}y^{6}\right)^{2}.

$$\frac{\left(-\frac{7}{4}\right)^{2}\left(x^{3}\right)^{2}\left(y^{6}\right)^{2}}{-14y}$$

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

$$\frac{\left(-\frac{7}{4}\right)^{2}x^{6}\left(y^{6}\right)^{2}}{-14y}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{\left(-\frac{7}{4}\right)^{2}x^{6}y^{12}}{-14y}$$

Calculate -\frac{7}{4} to the power of 2 and get \frac{49}{16}.

$$\frac{\frac{49}{16}x^{6}y^{12}}{-14y}$$

Cancel out y in both numerator and denominator.

$$\frac{\frac{49}{16}x^{6}y^{11}}{-14}$$

Divide \frac{49}{16}x^{6}y^{11} by -14 to get -\frac{7}{32}x^{6}y^{11}.

$$-\frac{7}{32}x^{6}y^{11}$$

Expand

$$-\frac{7x^{6}y^{11}}{32}$$

Solution Steps

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{\left(\frac{2}{3}\right)^{2}x^{2}y^{2}}{\frac{1}{3}x}\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{\frac{4}{9}x^{2}y^{2}}{\frac{1}{3}x}\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{\frac{4}{9}xy^{2}}{\frac{1}{3}}\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

Divide \frac{4}{9}xy^{2} by \frac{1}{3} by multiplying \frac{4}{9}xy^{2} by the reciprocal of \frac{1}{3}.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\left(\frac{3}{2}xy^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\left(\frac{3}{2}\right)^{2}x^{2}\left(y^{2}\right)^{2}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\left(\frac{3}{2}\right)^{2}x^{2}y^{4}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{9}{4}x^{2}y^{4}\times \frac{1}{3}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

Multiply \frac{9}{4} and \frac{1}{3} to get \frac{3}{4}.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{2}y^{4}xy^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{4}y^{2}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}xy^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}\right)^{-1}x^{-1}\left(y^{2}\right)^{-1}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-\frac{1}{2}\right)^{-1}x^{-1}y^{-2}}$$

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

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{7xy^{3}\left(-2\right)x^{-1}y^{-2}}$$

Multiply 7 and -2 to get -14.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14xy^{3}x^{-1}y^{-2}}$$

Multiply x and x^{-1} to get 1.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y^{3}y^{-2}}$$

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{9}xy^{2}\times 3\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y^{1}}$$

Multiply \frac{4}{9} and 3 to get \frac{4}{3}.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{3}xy^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y^{1}}$$

Calculate y to the power of 1 and get y.

$$\frac{\left(\left(\frac{1}{3}xy^{2}-\frac{4}{3}xy^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Combine \frac{1}{3}xy^{2} and -\frac{4}{3}xy^{2} to get -xy^{2}.

$$\frac{\left(\left(-xy^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

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

$$\frac{\left(\left(-1\right)^{3}x^{3}\left(y^{2}\right)^{3}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

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

$$\frac{\left(\left(-1\right)^{3}x^{3}y^{6}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Calculate -1 to the power of 3 and get -1.

$$\frac{\left(-x^{3}y^{6}-\frac{3}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Combine -x^{3}y^{6} and -\frac{3}{4}x^{3}y^{6} to get -\frac{7}{4}x^{3}y^{6}.

$$\frac{\left(-\frac{7}{4}x^{3}y^{6}\right)^{2}}{-14y}$$

Expand \left(-\frac{7}{4}x^{3}y^{6}\right)^{2}.

$$\frac{\left(-\frac{7}{4}\right)^{2}\left(x^{3}\right)^{2}\left(y^{6}\right)^{2}}{-14y}$$

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

$$\frac{\left(-\frac{7}{4}\right)^{2}x^{6}\left(y^{6}\right)^{2}}{-14y}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{\left(-\frac{7}{4}\right)^{2}x^{6}y^{12}}{-14y}$$

Calculate -\frac{7}{4} to the power of 2 and get \frac{49}{16}.

$$\frac{\frac{49}{16}x^{6}y^{12}}{-14y}$$

Cancel out y in both numerator and denominator.

$$\frac{\frac{49}{16}x^{6}y^{11}}{-14}$$

Divide \frac{49}{16}x^{6}y^{11} by -14 to get -\frac{7}{32}x^{6}y^{11}.

$$-\frac{7}{32}x^{6}y^{11}$$