Solve (1/10x^{2y^{-5}})^-2(20/x^-23y^12)^-1

$$( \frac { 1 } { 10 x ^ { 2 } y ^ { - 5 } } ) ^ { - 2 } ( \frac { 20 } { x ^ { - 23 } y ^ { 12 } } ) ^ { - 1 }$$

Evaluate

$$\frac{5y^{2}}{x^{19}}$$

Solution Steps

To raise \frac{1}{10x^{2}y^{-5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-2}}{\left(10x^{2}y^{-5}\right)^{-2}}\times \left(\frac{20}{x^{-23}y^{12}}\right)^{-1}$$

To raise \frac{20}{x^{-23}y^{12}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-2}}{\left(10x^{2}y^{-5}\right)^{-2}}\times \frac{20^{-1}}{\left(x^{-23}y^{12}\right)^{-1}}$$

Multiply \frac{1^{-2}}{\left(10x^{2}y^{-5}\right)^{-2}} times \frac{20^{-1}}{\left(x^{-23}y^{12}\right)^{-1}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1^{-2}\times 20^{-1}}{\left(10x^{2}y^{-5}\right)^{-2}\left(x^{-23}y^{12}\right)^{-1}}$$

Calculate 1 to the power of -2 and get 1.

$$\frac{1\times 20^{-1}}{\left(10x^{2}y^{-5}\right)^{-2}\left(x^{-23}y^{12}\right)^{-1}}$$

Calculate 20 to the power of -1 and get \frac{1}{20}.

$$\frac{1\times \frac{1}{20}}{\left(10x^{2}y^{-5}\right)^{-2}\left(x^{-23}y^{12}\right)^{-1}}$$

Multiply 1 and \frac{1}{20} to get \frac{1}{20}.

$$\frac{\frac{1}{20}}{\left(10x^{2}y^{-5}\right)^{-2}\left(x^{-23}y^{12}\right)^{-1}}$$

Expand \left(10x^{2}y^{-5}\right)^{-2}.

$$\frac{\frac{1}{20}}{10^{-2}\left(x^{2}\right)^{-2}\left(y^{-5}\right)^{-2}\left(x^{-23}y^{12}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -2 to get -4.

$$\frac{\frac{1}{20}}{10^{-2}x^{-4}\left(y^{-5}\right)^{-2}\left(x^{-23}y^{12}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -2 to get 10.

$$\frac{\frac{1}{20}}{10^{-2}x^{-4}y^{10}\left(x^{-23}y^{12}\right)^{-1}}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{\frac{1}{20}}{\frac{1}{100}x^{-4}y^{10}\left(x^{-23}y^{12}\right)^{-1}}$$

Expand \left(x^{-23}y^{12}\right)^{-1}.

$$\frac{\frac{1}{20}}{\frac{1}{100}x^{-4}y^{10}\left(x^{-23}\right)^{-1}\left(y^{12}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -23 and -1 to get 23.

$$\frac{\frac{1}{20}}{\frac{1}{100}x^{-4}y^{10}x^{23}\left(y^{12}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 12 and -1 to get -12.

$$\frac{\frac{1}{20}}{\frac{1}{100}x^{-4}y^{10}x^{23}y^{-12}}$$

To multiply powers of the same base, add their exponents. Add -4 and 23 to get 19.

$$\frac{\frac{1}{20}}{\frac{1}{100}x^{19}y^{10}y^{-12}}$$

To multiply powers of the same base, add their exponents. Add 10 and -12 to get -2.

$$\frac{\frac{1}{20}}{\frac{1}{100}x^{19}y^{-2}}$$

Express \frac{\frac{1}{20}}{\frac{1}{100}x^{19}y^{-2}} as a single fraction.

$$\frac{1}{20\times \frac{1}{100}x^{19}y^{-2}}$$

Multiply 20 and \frac{1}{100} to get \frac{1}{5}.

$$\frac{1}{\frac{1}{5}x^{19}y^{-2}}$$

Differentiate w.r.t. x

$$-\frac{95y^{2}}{x^{20}}$$