Solve frac{(2xy^{2}z^{-1})^{3}}{(4x^{3}y)^{0}(-6xy^-1z)^-2}div(-3x^2y)^3

$$\frac { ( 2 x y ^ { 2 } z ^ { - 1 } ) ^ { 3 } } { ( 4 x ^ { 3 } y ) ^ { 0 } ( - 6 x y ^ { - 1 } z ) ^ { - 2 } } \div ( - 3 x ^ { 2 } y ) ^ { 3 }$$

Evaluate

$$-\frac{32y}{3xz}$$

Solution Steps

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

$$\frac{\frac{2^{3}x^{3}\left(y^{2}\right)^{3}\left(z^{-1}\right)^{3}}{\left(4x^{3}y\right)^{0}\left(-6xy^{-1}z\right)^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

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

$$\frac{\frac{2^{3}x^{3}y^{6}\left(z^{-1}\right)^{3}}{\left(4x^{3}y\right)^{0}\left(-6xy^{-1}z\right)^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 3 to get -3.

$$\frac{\frac{2^{3}x^{3}y^{6}z^{-3}}{\left(4x^{3}y\right)^{0}\left(-6xy^{-1}z\right)^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\frac{8x^{3}y^{6}z^{-3}}{\left(4x^{3}y\right)^{0}\left(-6xy^{-1}z\right)^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

Calculate 4x^{3}y to the power of 0 and get 1.

$$\frac{\frac{8x^{3}y^{6}z^{-3}}{1\left(-6xy^{-1}z\right)^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

Expand \left(-6xy^{-1}z\right)^{-2}.

$$\frac{\frac{8x^{3}y^{6}z^{-3}}{1\left(-6\right)^{-2}x^{-2}\left(y^{-1}\right)^{-2}z^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

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

$$\frac{\frac{8x^{3}y^{6}z^{-3}}{1\left(-6\right)^{-2}x^{-2}y^{2}z^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

Calculate -6 to the power of -2 and get \frac{1}{36}.

$$\frac{\frac{8x^{3}y^{6}z^{-3}}{1\times \frac{1}{36}x^{-2}y^{2}z^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

Multiply 1 and \frac{1}{36} to get \frac{1}{36}.

$$\frac{\frac{8x^{3}y^{6}z^{-3}}{\frac{1}{36}x^{-2}y^{2}z^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

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

$$\frac{\frac{8z^{-3}x^{3}y^{4}}{\frac{1}{36}x^{-2}z^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\frac{8z^{-3}y^{4}x^{5}}{\frac{1}{36}z^{-2}}}{\left(-3x^{2}y\right)^{3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\frac{8y^{4}x^{5}}{\frac{1}{36}z^{1}}}{\left(-3x^{2}y\right)^{3}}$$

Calculate z to the power of 1 and get z.

$$\frac{\frac{8y^{4}x^{5}}{\frac{1}{36}z}}{\left(-3x^{2}y\right)^{3}}$$

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

$$\frac{\frac{8y^{4}x^{5}}{\frac{1}{36}z}}{\left(-3\right)^{3}\left(x^{2}\right)^{3}y^{3}}$$

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

$$\frac{\frac{8y^{4}x^{5}}{\frac{1}{36}z}}{\left(-3\right)^{3}x^{6}y^{3}}$$

Calculate -3 to the power of 3 and get -27.

$$\frac{\frac{8y^{4}x^{5}}{\frac{1}{36}z}}{-27x^{6}y^{3}}$$

Express \frac{\frac{8y^{4}x^{5}}{\frac{1}{36}z}}{-27x^{6}y^{3}} as a single fraction.

$$\frac{8y^{4}x^{5}}{\frac{1}{36}z\left(-27\right)x^{6}y^{3}}$$

Cancel out y^{3}x^{5} in both numerator and denominator.

$$\frac{8y}{-27\times \frac{1}{36}xz}$$

Multiply -27 and \frac{1}{36} to get -\frac{3}{4}.

$$\frac{8y}{-\frac{3}{4}xz}$$

Differentiate w.r.t. x

$$\frac{32y}{3zx^{2}}$$