Solve (frac{3x^{-2}y^4}{5x^4y^-2})^-1

$$( \frac { 3 x ^ { - 2 } y ^ { 4 } } { 5 x ^ { 4 } y ^ { - 2 } } ) ^ { - 1 }$$

Evaluate

$$\frac{5\times \left(\frac{x}{y}\right)^{6}}{3}$$

Solution Steps

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

$$\left(\frac{3x^{-2}y^{6}}{5x^{4}}\right)^{-1}$$

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

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

To raise \frac{3y^{6}}{5x^{6}} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(3y^{6}\right)^{-1}.

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

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

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

Calculate 3 to the power of -1 and get \frac{1}{3}.

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

Expand \left(5x^{6}\right)^{-1}.

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

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

$$\frac{\frac{1}{3}y^{-6}}{5^{-1}x^{-6}}$$

Calculate 5 to the power of -1 and get \frac{1}{5}.

$$\frac{\frac{1}{3}y^{-6}}{\frac{1}{5}x^{-6}}$$

Differentiate w.r.t. x

$$\frac{10x^{5}}{y^{6}}$$