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

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

Evaluate

$$\frac{x^{4}y^{5}}{9}$$

Solution Steps

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

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

To raise \frac{3x^{-3}}{y^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Multiply \frac{\left(y^{-1}\right)^{-1}}{\left(x^{-2}\right)^{-1}} times \frac{\left(3x^{-3}\right)^{-2}}{\left(y^{2}\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

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

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

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

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

Expand \left(3x^{-3}\right)^{-2}.

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

To raise a power to another power, multiply the exponents. Multiply -3 and -2 to get 6.

$$\frac{y^{5}\times 3^{-2}x^{6}}{x^{2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{y^{5}\times \frac{1}{9}x^{6}}{x^{2}}$$

Cancel out x^{2} in both numerator and denominator.

$$\frac{1}{9}x^{4}y^{5}$$

Expand

$$\frac{x^{4}y^{5}}{9}$$

Solution Steps

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

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

To raise \frac{3x^{-3}}{y^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Multiply \frac{\left(y^{-1}\right)^{-1}}{\left(x^{-2}\right)^{-1}} times \frac{\left(3x^{-3}\right)^{-2}}{\left(y^{2}\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

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

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

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

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

Expand \left(3x^{-3}\right)^{-2}.

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

To raise a power to another power, multiply the exponents. Multiply -3 and -2 to get 6.

$$\frac{y^{5}\times 3^{-2}x^{6}}{x^{2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{y^{5}\times \frac{1}{9}x^{6}}{x^{2}}$$

Cancel out x^{2} in both numerator and denominator.

$$\frac{1}{9}x^{4}y^{5}$$