Solve (frac{xy^{-2}z^{-3}}{x^2y^3z^-4})^-3

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

Evaluate

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

Solution Steps

Cancel out x in both numerator and denominator.

$$\left(\frac{z^{-3}y^{-2}}{z^{-4}xy^{3}}\right)^{-3}$$

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

$$\left(\frac{y^{-2}z^{1}}{xy^{3}}\right)^{-3}$$

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

$$\left(\frac{z^{1}}{xy^{5}}\right)^{-3}$$

Calculate z to the power of 1 and get z.

$$\left(\frac{z}{xy^{5}}\right)^{-3}$$

To raise \frac{z}{xy^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{z^{-3}}{\left(xy^{5}\right)^{-3}}$$

Expand \left(xy^{5}\right)^{-3}.

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

To raise a power to another power, multiply the exponents. Multiply 5 and -3 to get -15.

$$\frac{z^{-3}}{x^{-3}y^{-15}}$$

Expand

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

Solution Steps

Cancel out x in both numerator and denominator.

$$\left(\frac{z^{-3}y^{-2}}{z^{-4}xy^{3}}\right)^{-3}$$

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

$$\left(\frac{y^{-2}z^{1}}{xy^{3}}\right)^{-3}$$

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

$$\left(\frac{z^{1}}{xy^{5}}\right)^{-3}$$

Calculate z to the power of 1 and get z.

$$\left(\frac{z}{xy^{5}}\right)^{-3}$$

To raise \frac{z}{xy^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{z^{-3}}{\left(xy^{5}\right)^{-3}}$$

Expand \left(xy^{5}\right)^{-3}.

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

To raise a power to another power, multiply the exponents. Multiply 5 and -3 to get -15.

$$\frac{z^{-3}}{x^{-3}y^{-15}}$$