Solve (frac{x^{-1}y^{5}z^-2}{x^4y^7z})^-1

$$( \frac { x ^ { - 1 } y ^ { 5 } z ^ { - 2 } } { x ^ { 4 } y ^ { 7 } z } ) ^ { - 1 }$$

Evaluate

$$y^{2}z^{3}x^{5}$$

Solution Steps

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

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

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

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

To raise \frac{1}{y^{2}z^{3}x^{5}} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 1 to the power of -1 and get 1.

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

Expand \left(y^{2}z^{3}x^{5}\right)^{-1}.

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 5 and -1 to get -5.

$$\frac{1}{y^{-2}z^{-3}x^{-5}}$$

Differentiate w.r.t. x

$$5y^{2}z^{3}x^{4}$$