Solve (-1/2xy^{2}z-1/4xy^{2}z)^{2}(x^{2}yz-1/3x^{2}yz)^{2}:(x^{4}y^{3}z^{2}-1/6x^{4}y^{2}z^{2})]^{2}:(-frac{1}{5}xy^{2}z^2)^2-frac{1}{2}x^2y^2

$$( - \frac { 1 } { 2 } x y ^ { 2 } z - \frac { 1 } { 4 } x y ^ { 2 } z ) ^ { 2 } ( x ^ { 2 } y z - \frac { 1 } { 3 } x ^ { 2 } y z ) ^ { 2 } : ( x ^ { 4 } y ^ { 3 } z ^ { 2 } - \frac { 1 } { 6 } x ^ { 4 } y ^ { 2 } z ^ { 2 } ) ] ^ { 2 } : ( - \frac { 1 } { 5 } x y ^ { 2 } z ^ { 2 } ) ^ { 2 } - \frac { 1 } { 2 } x ^ { 2 } y ^ { 2 }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\left(x^{2}yz-\frac{1}{3}x^{2}yz\right)^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

Combine x^{2}yz and -\frac{1}{3}x^{2}yz to get \frac{2}{3}x^{2}yz.

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \left(\frac{2}{3}x^{2}yz\right)^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \left(\frac{2}{3}\right)^{2}\left(x^{2}\right)^{2}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \left(\frac{2}{3}\right)^{2}x^{4}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \frac{4}{9}x^{4}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

Multiply \frac{9}{16} and \frac{4}{9} to get \frac{1}{4}.

$$\frac{\frac{1}{4}x^{2}y^{4}z^{2}x^{4}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{1}{4}x^{6}y^{4}z^{2}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{1}{4}x^{6}y^{6}z^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{1}{4}x^{6}y^{6}z^{4}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Expand \left(xz\right)^{2}.

$$\frac{\frac{1}{4}y^{4}x^{2}z^{2}}{y-\frac{1}{6}}$$

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\left(x^{2}yz-\frac{1}{3}x^{2}yz\right)^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

Combine x^{2}yz and -\frac{1}{3}x^{2}yz to get \frac{2}{3}x^{2}yz.

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \left(\frac{2}{3}x^{2}yz\right)^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \left(\frac{2}{3}\right)^{2}\left(x^{2}\right)^{2}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \left(\frac{2}{3}\right)^{2}x^{4}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{9}{16}x^{2}y^{4}z^{2}\times \frac{4}{9}x^{4}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

Multiply \frac{9}{16} and \frac{4}{9} to get \frac{1}{4}.

$$\frac{\frac{1}{4}x^{2}y^{4}z^{2}x^{4}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{1}{4}x^{6}y^{4}z^{2}y^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{1}{4}x^{6}y^{6}z^{2}z^{2}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

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

$$\frac{\frac{1}{4}x^{6}y^{6}z^{4}}{x^{4}y^{3}z^{2}-\frac{1}{6}x^{4}y^{2}z^{2}}$$

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Expand \left(xz\right)^{2}.

$$\frac{\frac{1}{4}y^{4}x^{2}z^{2}}{y-\frac{1}{6}}$$