Solve (2/3y^{2}z)^{5}:(2/3y^{2}z)^{3}-[-3/20y^5z^2-(+11/4y^5z^2)]:[-(5y+frac{4}{5}y)]

$$( \frac { 2 } { 3 } y ^ { 2 } z ) ^ { 5 } : ( \frac { 2 } { 3 } y ^ { 2 } z ) ^ { 3 } - [ - \frac { 3 } { 20 } y ^ { 5 } z ^ { 2 } - ( + \frac { 11 } { 4 } y ^ { 5 } z ^ { 2 } ) ] : [ - ( 5 y + \frac { 4 } { 5 } y ) ]$$

Evaluate

$$-\frac{z^{2}y^{4}}{18}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 3 from 5 to get 2.

$$\left(\frac{2}{3}y^{2}z\right)^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

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

$$\left(\frac{2}{3}\right)^{2}\left(y^{2}\right)^{2}z^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

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

$$\left(\frac{2}{3}\right)^{2}y^{4}z^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

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

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

Combine -\frac{3}{20}y^{5}z^{2} and -\frac{11}{4}y^{5}z^{2} to get -\frac{29}{10}y^{5}z^{2}.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

Combine 5y and \frac{4}{5}y to get \frac{29}{5}y.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}y^{5}z^{2}}{-\frac{29}{5}y}$$

Cancel out y in both numerator and denominator.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}z^{2}y^{4}}{-\frac{29}{5}}$$

Divide -\frac{29}{10}z^{2}y^{4} by -\frac{29}{5} by multiplying -\frac{29}{10}z^{2}y^{4} by the reciprocal of -\frac{29}{5}.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}z^{2}y^{4}\times 5}{-29}$$

Multiply -\frac{29}{10} and 5 to get -\frac{29}{2}.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{2}z^{2}y^{4}}{-29}$$

Divide -\frac{29}{2}z^{2}y^{4} by -29 to get \frac{1}{2}z^{2}y^{4}.

$$\frac{4}{9}y^{4}z^{2}-\frac{1}{2}z^{2}y^{4}$$

Combine \frac{4}{9}y^{4}z^{2} and -\frac{1}{2}z^{2}y^{4} to get -\frac{1}{18}y^{4}z^{2}.

$$-\frac{1}{18}y^{4}z^{2}$$

Expand

$$-\frac{z^{2}y^{4}}{18}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 3 from 5 to get 2.

$$\left(\frac{2}{3}y^{2}z\right)^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

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

$$\left(\frac{2}{3}\right)^{2}\left(y^{2}\right)^{2}z^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

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

$$\left(\frac{2}{3}\right)^{2}y^{4}z^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

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

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{3}{20}y^{5}z^{2}-\frac{11}{4}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

Combine -\frac{3}{20}y^{5}z^{2} and -\frac{11}{4}y^{5}z^{2} to get -\frac{29}{10}y^{5}z^{2}.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}y^{5}z^{2}}{-\left(5y+\frac{4}{5}y\right)}$$

Combine 5y and \frac{4}{5}y to get \frac{29}{5}y.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}y^{5}z^{2}}{-\frac{29}{5}y}$$

Cancel out y in both numerator and denominator.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}z^{2}y^{4}}{-\frac{29}{5}}$$

Divide -\frac{29}{10}z^{2}y^{4} by -\frac{29}{5} by multiplying -\frac{29}{10}z^{2}y^{4} by the reciprocal of -\frac{29}{5}.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{10}z^{2}y^{4}\times 5}{-29}$$

Multiply -\frac{29}{10} and 5 to get -\frac{29}{2}.

$$\frac{4}{9}y^{4}z^{2}-\frac{-\frac{29}{2}z^{2}y^{4}}{-29}$$

Divide -\frac{29}{2}z^{2}y^{4} by -29 to get \frac{1}{2}z^{2}y^{4}.

$$\frac{4}{9}y^{4}z^{2}-\frac{1}{2}z^{2}y^{4}$$

Combine \frac{4}{9}y^{4}z^{2} and -\frac{1}{2}z^{2}y^{4} to get -\frac{1}{18}y^{4}z^{2}.

$$-\frac{1}{18}y^{4}z^{2}$$