Solve (-3x^{2}y^{3})(x^-2y^-3/x^4y)^-1

$$( - 3 x ^ { 2 } y ^ { 3 } ) ( \frac { x ^ { - 2 } y ^ { - 3 } } { x ^ { 4 } y } ) ^ { - 1 }$$

Evaluate

$$-3y^{7}x^{8}$$

Solution Steps

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

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

To raise \frac{1}{y^{4}x^{6}} to a power, raise both numerator and denominator to the power and then divide.

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

Express -3\times \frac{1^{-1}}{\left(y^{4}x^{6}\right)^{-1}} as a single fraction.

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

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

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

Expand \left(y^{4}x^{6}\right)^{-1}.

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

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

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

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

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

Express \frac{-3}{y^{-4}x^{-6}}x^{2} as a single fraction.

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

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

$$\frac{-3x^{8}}{y^{-4}}y^{3}$$

Express \frac{-3x^{8}}{y^{-4}}y^{3} as a single fraction.

$$\frac{-3x^{8}y^{3}}{y^{-4}}$$

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

$$-3y^{7}x^{8}$$

Differentiate w.r.t. x

$$-24\left(xy\right)^{7}$$