Solve 6*(frac{x^{3}y^{2}z^{4}}{x^-4y^3z^-2})^-1=

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

Evaluate

$$\frac{6y}{z^{6}x^{7}}$$

Solution Steps

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

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

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

$$6\times \left(\frac{z^{6}x^{7}}{y}\right)^{-1}$$

To raise \frac{z^{6}x^{7}}{y} to a power, raise both numerator and denominator to the power and then divide.

$$6\times \frac{\left(z^{6}x^{7}\right)^{-1}}{y^{-1}}$$

Express 6\times \frac{\left(z^{6}x^{7}\right)^{-1}}{y^{-1}} as a single fraction.

$$\frac{6\left(z^{6}x^{7}\right)^{-1}}{y^{-1}}$$

Expand \left(z^{6}x^{7}\right)^{-1}.

$$\frac{6\left(z^{6}\right)^{-1}\left(x^{7}\right)^{-1}}{y^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -1 to get -6.

$$\frac{6z^{-6}\left(x^{7}\right)^{-1}}{y^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -1 to get -7.

$$\frac{6z^{-6}x^{-7}}{y^{-1}}$$

Differentiate w.r.t. x

$$-\frac{42y}{z^{6}x^{8}}$$