Solve (frac{27x^{-5}y^{-3}}{3^5x^-7y^-5})^-1

$$( \frac { 27 x ^ { - 5 } y ^ { - 3 } } { 3 ^ { 5 } x ^ { - 7 } y ^ { - 5 } } ) ^ { - 1 }$$

Evaluate

$$\frac{9}{\left(xy\right)^{2}}$$

Solution Steps

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

$$\left(\frac{27x^{2}y^{2}}{3^{5}}\right)^{-1}$$

Calculate 3 to the power of 5 and get 243.

$$\left(\frac{27x^{2}y^{2}}{243}\right)^{-1}$$

Divide 27x^{2}y^{2} by 243 to get \frac{1}{9}x^{2}y^{2}.

$$\left(\frac{1}{9}x^{2}y^{2}\right)^{-1}$$

Expand \left(\frac{1}{9}x^{2}y^{2}\right)^{-1}.

$$\left(\frac{1}{9}\right)^{-1}\left(x^{2}\right)^{-1}\left(y^{2}\right)^{-1}$$

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

$$\left(\frac{1}{9}\right)^{-1}x^{-2}\left(y^{2}\right)^{-1}$$

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

$$\left(\frac{1}{9}\right)^{-1}x^{-2}y^{-2}$$

Calculate \frac{1}{9} to the power of -1 and get 9.

$$9x^{-2}y^{-2}$$

Differentiate w.r.t. x

$$-\frac{18}{y^{2}x^{3}}$$