Solve (-3/4yz^{2})^{3}(+3/4yz^{2}):(+3/4yz^{2})^{2}+[3/8y^{2}(frac{3}{5}z^{3}-frac{1}{3}z^3):(-frac{2}{5}yz)]^2-frac{7}{8}y^2z^4

$$( - \frac { 3 } { 4 } y z ^ { 2 } ) ^ { 3 } ( + \frac { 3 } { 4 } y z ^ { 2 } ) : ( + \frac { 3 } { 4 } y z ^ { 2 } ) ^ { 2 } + [ \frac { 3 } { 8 } y ^ { 2 } ( \frac { 3 } { 5 } z ^ { 3 } - \frac { 1 } { 3 } z ^ { 3 } ) : ( - \frac { 2 } { 5 } y z ) ] ^ { 2 } - \frac { 7 } { 8 } y ^ { 2 } z ^ { 4 }$$

Evaluate

$$-\frac{11y^{2}z^{4}}{8}$$

Solution Steps

Expand \left(-\frac{3}{4}yz^{2}\right)^{3}.

$$\frac{\left(-\frac{3}{4}\right)^{3}y^{3}\left(z^{2}\right)^{3}\times \frac{3}{4}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{\left(-\frac{3}{4}\right)^{3}y^{3}z^{6}\times \frac{3}{4}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Calculate -\frac{3}{4} to the power of 3 and get -\frac{27}{64}.

$$\frac{-\frac{27}{64}y^{3}z^{6}\times \frac{3}{4}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply -\frac{27}{64} and \frac{3}{4} to get -\frac{81}{256}.

$$\frac{-\frac{81}{256}y^{3}z^{6}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{-\frac{81}{256}y^{4}z^{6}z^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Expand \left(\frac{3}{4}yz^{2}\right)^{2}.

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\left(\frac{3}{4}\right)^{2}y^{2}\left(z^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\left(\frac{3}{4}\right)^{2}y^{2}z^{4}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Calculate \frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\frac{9}{16}y^{2}z^{4}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Cancel out y^{2}z^{4} in both numerator and denominator.

$$\frac{-\frac{81}{256}y^{2}z^{4}}{\frac{9}{16}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide -\frac{81}{256}y^{2}z^{4} by \frac{9}{16} by multiplying -\frac{81}{256}y^{2}z^{4} by the reciprocal of \frac{9}{16}.

$$\frac{-\frac{81}{256}y^{2}z^{4}\times 16}{9}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply -\frac{81}{256} and 16 to get -\frac{81}{16}.

$$\frac{-\frac{81}{16}y^{2}z^{4}}{9}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide -\frac{81}{16}y^{2}z^{4} by 9 to get -\frac{9}{16}y^{2}z^{4}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Cancel out y in both numerator and denominator.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{3}{8}y\left(-\frac{1}{3}z^{3}+\frac{3}{5}z^{3}\right)}{-\frac{2}{5}z}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Combine -\frac{1}{3}z^{3} and \frac{3}{5}z^{3} to get \frac{4}{15}z^{3}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{3}{8}y\times \frac{4}{15}z^{3}}{-\frac{2}{5}z}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply \frac{3}{8} and \frac{4}{15} to get \frac{1}{10}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{10}yz^{3}}{-\frac{2}{5}z}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Cancel out z in both numerator and denominator.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{10}yz^{2}}{-\frac{2}{5}}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide \frac{1}{10}yz^{2} by -\frac{2}{5} by multiplying \frac{1}{10}yz^{2} by the reciprocal of -\frac{2}{5}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{10}yz^{2}\times 5}{-2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply \frac{1}{10} and 5 to get \frac{1}{2}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{2}yz^{2}}{-2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide \frac{1}{2}yz^{2} by -2 to get -\frac{1}{4}yz^{2}.

$$-\frac{9}{16}y^{2}z^{4}+\left(-\frac{1}{4}yz^{2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Expand \left(-\frac{1}{4}yz^{2}\right)^{2}.

$$-\frac{9}{16}y^{2}z^{4}+\left(-\frac{1}{4}\right)^{2}y^{2}\left(z^{2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$-\frac{9}{16}y^{2}z^{4}+\left(-\frac{1}{4}\right)^{2}y^{2}z^{4}-\frac{7}{8}y^{2}z^{4}$$

Calculate -\frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$-\frac{9}{16}y^{2}z^{4}+\frac{1}{16}y^{2}z^{4}-\frac{7}{8}y^{2}z^{4}$$

Combine -\frac{9}{16}y^{2}z^{4} and \frac{1}{16}y^{2}z^{4} to get -\frac{1}{2}y^{2}z^{4}.

$$-\frac{1}{2}y^{2}z^{4}-\frac{7}{8}y^{2}z^{4}$$

Combine -\frac{1}{2}y^{2}z^{4} and -\frac{7}{8}y^{2}z^{4} to get -\frac{11}{8}y^{2}z^{4}.

$$-\frac{11}{8}y^{2}z^{4}$$

Expand

$$-\frac{11y^{2}z^{4}}{8}$$

Solution Steps

Expand \left(-\frac{3}{4}yz^{2}\right)^{3}.

$$\frac{\left(-\frac{3}{4}\right)^{3}y^{3}\left(z^{2}\right)^{3}\times \frac{3}{4}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{\left(-\frac{3}{4}\right)^{3}y^{3}z^{6}\times \frac{3}{4}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Calculate -\frac{3}{4} to the power of 3 and get -\frac{27}{64}.

$$\frac{-\frac{27}{64}y^{3}z^{6}\times \frac{3}{4}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply -\frac{27}{64} and \frac{3}{4} to get -\frac{81}{256}.

$$\frac{-\frac{81}{256}y^{3}z^{6}yz^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{-\frac{81}{256}y^{4}z^{6}z^{2}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\left(\frac{3}{4}yz^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Expand \left(\frac{3}{4}yz^{2}\right)^{2}.

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\left(\frac{3}{4}\right)^{2}y^{2}\left(z^{2}\right)^{2}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\left(\frac{3}{4}\right)^{2}y^{2}z^{4}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Calculate \frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\frac{-\frac{81}{256}y^{4}z^{8}}{\frac{9}{16}y^{2}z^{4}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Cancel out y^{2}z^{4} in both numerator and denominator.

$$\frac{-\frac{81}{256}y^{2}z^{4}}{\frac{9}{16}}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide -\frac{81}{256}y^{2}z^{4} by \frac{9}{16} by multiplying -\frac{81}{256}y^{2}z^{4} by the reciprocal of \frac{9}{16}.

$$\frac{-\frac{81}{256}y^{2}z^{4}\times 16}{9}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply -\frac{81}{256} and 16 to get -\frac{81}{16}.

$$\frac{-\frac{81}{16}y^{2}z^{4}}{9}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide -\frac{81}{16}y^{2}z^{4} by 9 to get -\frac{9}{16}y^{2}z^{4}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{3}{8}y^{2}\left(\frac{3}{5}z^{3}-\frac{1}{3}z^{3}\right)}{-\frac{2}{5}yz}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Cancel out y in both numerator and denominator.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{3}{8}y\left(-\frac{1}{3}z^{3}+\frac{3}{5}z^{3}\right)}{-\frac{2}{5}z}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Combine -\frac{1}{3}z^{3} and \frac{3}{5}z^{3} to get \frac{4}{15}z^{3}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{3}{8}y\times \frac{4}{15}z^{3}}{-\frac{2}{5}z}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply \frac{3}{8} and \frac{4}{15} to get \frac{1}{10}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{10}yz^{3}}{-\frac{2}{5}z}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Cancel out z in both numerator and denominator.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{10}yz^{2}}{-\frac{2}{5}}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide \frac{1}{10}yz^{2} by -\frac{2}{5} by multiplying \frac{1}{10}yz^{2} by the reciprocal of -\frac{2}{5}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{10}yz^{2}\times 5}{-2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Multiply \frac{1}{10} and 5 to get \frac{1}{2}.

$$-\frac{9}{16}y^{2}z^{4}+\left(\frac{\frac{1}{2}yz^{2}}{-2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Divide \frac{1}{2}yz^{2} by -2 to get -\frac{1}{4}yz^{2}.

$$-\frac{9}{16}y^{2}z^{4}+\left(-\frac{1}{4}yz^{2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

Expand \left(-\frac{1}{4}yz^{2}\right)^{2}.

$$-\frac{9}{16}y^{2}z^{4}+\left(-\frac{1}{4}\right)^{2}y^{2}\left(z^{2}\right)^{2}-\frac{7}{8}y^{2}z^{4}$$

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

$$-\frac{9}{16}y^{2}z^{4}+\left(-\frac{1}{4}\right)^{2}y^{2}z^{4}-\frac{7}{8}y^{2}z^{4}$$

Calculate -\frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$-\frac{9}{16}y^{2}z^{4}+\frac{1}{16}y^{2}z^{4}-\frac{7}{8}y^{2}z^{4}$$

Combine -\frac{9}{16}y^{2}z^{4} and \frac{1}{16}y^{2}z^{4} to get -\frac{1}{2}y^{2}z^{4}.

$$-\frac{1}{2}y^{2}z^{4}-\frac{7}{8}y^{2}z^{4}$$

Combine -\frac{1}{2}y^{2}z^{4} and -\frac{7}{8}y^{2}z^{4} to get -\frac{11}{8}y^{2}z^{4}.

$$-\frac{11}{8}y^{2}z^{4}$$