Solve [(x^-2)/(xy)^3]^15

$$[ \frac { ( x ^ { - 2 } ) } { ( x y ) ^ { 3 } } ] ^ { 15 }$$

Evaluate

$$\frac{1}{y^{45}x^{75}}$$

Solution Steps

Expand \left(xy\right)^{3}.

$$\left(\frac{x^{-2}}{x^{3}y^{3}}\right)^{15}$$

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

$$\left(\frac{1}{y^{3}x^{5}}\right)^{15}$$

To raise \frac{1}{y^{3}x^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{15}}{\left(y^{3}x^{5}\right)^{15}}$$

Calculate 1 to the power of 15 and get 1.

$$\frac{1}{\left(y^{3}x^{5}\right)^{15}}$$

Expand \left(y^{3}x^{5}\right)^{15}.

$$\frac{1}{\left(y^{3}\right)^{15}\left(x^{5}\right)^{15}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 15 to get 45.

$$\frac{1}{y^{45}\left(x^{5}\right)^{15}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 15 to get 75.

$$\frac{1}{y^{45}x^{75}}$$

Expand

$$\frac{1}{y^{45}x^{75}}$$

Solution Steps

Expand \left(xy\right)^{3}.

$$\left(\frac{x^{-2}}{x^{3}y^{3}}\right)^{15}$$

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

$$\left(\frac{1}{y^{3}x^{5}}\right)^{15}$$

To raise \frac{1}{y^{3}x^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{15}}{\left(y^{3}x^{5}\right)^{15}}$$

Calculate 1 to the power of 15 and get 1.

$$\frac{1}{\left(y^{3}x^{5}\right)^{15}}$$

Expand \left(y^{3}x^{5}\right)^{15}.

$$\frac{1}{\left(y^{3}\right)^{15}\left(x^{5}\right)^{15}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 15 to get 45.

$$\frac{1}{y^{45}\left(x^{5}\right)^{15}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 15 to get 75.

$$\frac{1}{y^{45}x^{75}}$$