Solve [-(-1/4xy^{2}z)^{3}-(1/2xy^{2})^{2}(-1/16xy^{2}z^{3})+3/4x^{3}y^{6}z^3]:(-frac{5}{4}xy^2z)^2quadfrac{1}{2}xy^2z

$$[ - ( - \frac { 1 } { 4 } x y ^ { 2 } z ) ^ { 3 } - ( \frac { 1 } { 2 } x y ^ { 2 } ) ^ { 2 } ( - \frac { 1 } { 16 } x y ^ { 2 } z ^ { 3 } ) + \frac { 3 } { 4 } x ^ { 3 } y ^ { 6 } z ^ { 3 } ] : ( - \frac { 5 } { 4 } x y ^ { 2 } z ) ^ { 2 } \quad \frac { 1 } { 2 } x y ^ { 2 } z$$

Evaluate

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

Solution Steps

Expand \left(-\frac{1}{4}xy^{2}z\right)^{3}.

$$\frac{-\left(-\frac{1}{4}\right)^{3}x^{3}\left(y^{2}\right)^{3}z^{3}-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{-\left(-\frac{1}{4}\right)^{3}x^{3}y^{6}z^{3}-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Calculate -\frac{1}{4} to the power of 3 and get -\frac{1}{64}.

$$\frac{-\left(-\frac{1}{64}x^{3}y^{6}z^{3}\right)-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

The opposite of -\frac{1}{64}x^{3}y^{6}z^{3} is \frac{1}{64}x^{3}y^{6}z^{3}.

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(\frac{1}{2}\right)^{2}x^{2}\left(y^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(\frac{1}{2}\right)^{2}x^{2}y^{4}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\frac{1}{4}x^{2}y^{4}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Multiply \frac{1}{4} and -\frac{1}{16} to get -\frac{1}{64}.

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(-\frac{1}{64}x^{2}y^{4}xy^{2}z^{3}\right)+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(-\frac{1}{64}x^{3}y^{4}y^{2}z^{3}\right)+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(-\frac{1}{64}x^{3}y^{6}z^{3}\right)+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

The opposite of -\frac{1}{64}x^{3}y^{6}z^{3} is \frac{1}{64}x^{3}y^{6}z^{3}.

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}+\frac{1}{64}x^{3}y^{6}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Combine \frac{1}{64}x^{3}y^{6}z^{3} and \frac{1}{64}x^{3}y^{6}z^{3} to get \frac{1}{32}x^{3}y^{6}z^{3}.

$$\frac{\frac{1}{32}x^{3}y^{6}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Combine \frac{1}{32}x^{3}y^{6}z^{3} and \frac{3}{4}x^{3}y^{6}z^{3} to get \frac{25}{32}x^{3}y^{6}z^{3}.

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Expand \left(-\frac{5}{4}xy^{2}z\right)^{2}.

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}\right)^{2}x^{2}\left(y^{2}\right)^{2}z^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}\right)^{2}x^{2}y^{4}z^{2}}\times \frac{1}{2}xy^{2}z$$

Calculate -\frac{5}{4} to the power of 2 and get \frac{25}{16}.

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\frac{25}{16}x^{2}y^{4}z^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{25}{32}xzy^{2}}{\frac{25}{16}}\times \frac{1}{2}xy^{2}z$$

Divide \frac{25}{32}xzy^{2} by \frac{25}{16} by multiplying \frac{25}{32}xzy^{2} by the reciprocal of \frac{25}{16}.

$$\frac{\frac{25}{32}xzy^{2}\times 16}{25}\times \frac{1}{2}xy^{2}z$$

Multiply \frac{25}{32} and 16 to get \frac{25}{2}.

$$\frac{\frac{25}{2}xzy^{2}}{25}\times \frac{1}{2}xy^{2}z$$

Divide \frac{25}{2}xzy^{2} by 25 to get \frac{1}{2}xzy^{2}.

$$\frac{1}{2}xzy^{2}\times \frac{1}{2}xy^{2}z$$

Multiply \frac{1}{2} and \frac{1}{2} to get \frac{1}{4}.

$$\frac{1}{4}xzy^{2}xy^{2}z$$

Multiply x and x to get x^{2}.

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

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

$$\frac{1}{4}x^{2}zy^{4}z$$

Multiply z and z to get z^{2}.

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

Expand

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

Solution Steps

Expand \left(-\frac{1}{4}xy^{2}z\right)^{3}.

$$\frac{-\left(-\frac{1}{4}\right)^{3}x^{3}\left(y^{2}\right)^{3}z^{3}-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{-\left(-\frac{1}{4}\right)^{3}x^{3}y^{6}z^{3}-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Calculate -\frac{1}{4} to the power of 3 and get -\frac{1}{64}.

$$\frac{-\left(-\frac{1}{64}x^{3}y^{6}z^{3}\right)-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

The opposite of -\frac{1}{64}x^{3}y^{6}z^{3} is \frac{1}{64}x^{3}y^{6}z^{3}.

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(\frac{1}{2}xy^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(\frac{1}{2}\right)^{2}x^{2}\left(y^{2}\right)^{2}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(\frac{1}{2}\right)^{2}x^{2}y^{4}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\frac{1}{4}x^{2}y^{4}\left(-\frac{1}{16}\right)xy^{2}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Multiply \frac{1}{4} and -\frac{1}{16} to get -\frac{1}{64}.

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(-\frac{1}{64}x^{2}y^{4}xy^{2}z^{3}\right)+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(-\frac{1}{64}x^{3}y^{4}y^{2}z^{3}\right)+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}-\left(-\frac{1}{64}x^{3}y^{6}z^{3}\right)+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

The opposite of -\frac{1}{64}x^{3}y^{6}z^{3} is \frac{1}{64}x^{3}y^{6}z^{3}.

$$\frac{\frac{1}{64}x^{3}y^{6}z^{3}+\frac{1}{64}x^{3}y^{6}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Combine \frac{1}{64}x^{3}y^{6}z^{3} and \frac{1}{64}x^{3}y^{6}z^{3} to get \frac{1}{32}x^{3}y^{6}z^{3}.

$$\frac{\frac{1}{32}x^{3}y^{6}z^{3}+\frac{3}{4}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Combine \frac{1}{32}x^{3}y^{6}z^{3} and \frac{3}{4}x^{3}y^{6}z^{3} to get \frac{25}{32}x^{3}y^{6}z^{3}.

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}xy^{2}z\right)^{2}}\times \frac{1}{2}xy^{2}z$$

Expand \left(-\frac{5}{4}xy^{2}z\right)^{2}.

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}\right)^{2}x^{2}\left(y^{2}\right)^{2}z^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\left(-\frac{5}{4}\right)^{2}x^{2}y^{4}z^{2}}\times \frac{1}{2}xy^{2}z$$

Calculate -\frac{5}{4} to the power of 2 and get \frac{25}{16}.

$$\frac{\frac{25}{32}x^{3}y^{6}z^{3}}{\frac{25}{16}x^{2}y^{4}z^{2}}\times \frac{1}{2}xy^{2}z$$

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

$$\frac{\frac{25}{32}xzy^{2}}{\frac{25}{16}}\times \frac{1}{2}xy^{2}z$$

Divide \frac{25}{32}xzy^{2} by \frac{25}{16} by multiplying \frac{25}{32}xzy^{2} by the reciprocal of \frac{25}{16}.

$$\frac{\frac{25}{32}xzy^{2}\times 16}{25}\times \frac{1}{2}xy^{2}z$$

Multiply \frac{25}{32} and 16 to get \frac{25}{2}.

$$\frac{\frac{25}{2}xzy^{2}}{25}\times \frac{1}{2}xy^{2}z$$

Divide \frac{25}{2}xzy^{2} by 25 to get \frac{1}{2}xzy^{2}.

$$\frac{1}{2}xzy^{2}\times \frac{1}{2}xy^{2}z$$

Multiply \frac{1}{2} and \frac{1}{2} to get \frac{1}{4}.

$$\frac{1}{4}xzy^{2}xy^{2}z$$

Multiply x and x to get x^{2}.

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

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

$$\frac{1}{4}x^{2}zy^{4}z$$

Multiply z and z to get z^{2}.

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