Solve (frac{6x^{3}y^-2}{12x^-2y^4})^-1

$$( \frac { 6 x ^ { 3 } y ^ { - 2 } } { 12 x ^ { - 2 } y ^ { 4 } } ) ^ { - 1 }$$

Evaluate

$$\frac{2y^{6}}{x^{5}}$$

Solution Steps

Cancel out 6 in both numerator and denominator.

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

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

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

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

$$\left(\frac{x^{5}}{2y^{6}}\right)^{-1}$$

To raise \frac{x^{5}}{2y^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{5}\right)^{-1}}{\left(2y^{6}\right)^{-1}}$$

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

$$\frac{x^{-5}}{\left(2y^{6}\right)^{-1}}$$

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

$$\frac{x^{-5}}{2^{-1}\left(y^{6}\right)^{-1}}$$

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

$$\frac{x^{-5}}{2^{-1}y^{-6}}$$

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

$$\frac{x^{-5}}{\frac{1}{2}y^{-6}}$$

Differentiate w.r.t. x

$$-10\times \left(\frac{y}{x}\right)^{6}$$