Solve (frac{xy^{-2}z^{-3}}{x^{2}y^{3}z^-4})^-3div(xy/z^-1)^2

$$( \frac { x y ^ { - 2 } z ^ { - 3 } } { x ^ { 2 } y ^ { 3 } z ^ { - 4 } } ) ^ { - 3 } \div ( \frac { x y } { z ^ { - 1 } } ) ^ { 2 }$$

Evaluate

$$\frac{xy^{13}}{z^{5}}$$

Solution Steps

Cancel out x in both numerator and denominator.

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

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

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

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

$$\frac{\left(\frac{z^{1}}{xy^{5}}\right)^{-3}}{\left(\frac{xy}{z^{-1}}\right)^{2}}$$

Calculate z to the power of 1 and get z.

$$\frac{\left(\frac{z}{xy^{5}}\right)^{-3}}{\left(\frac{xy}{z^{-1}}\right)^{2}}$$

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

$$\frac{\frac{z^{-3}}{\left(xy^{5}\right)^{-3}}}{\left(\frac{xy}{z^{-1}}\right)^{2}}$$

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

$$\frac{\frac{z^{-3}}{\left(xy^{5}\right)^{-3}}}{\frac{\left(xy\right)^{2}}{\left(z^{-1}\right)^{2}}}$$

Divide \frac{z^{-3}}{\left(xy^{5}\right)^{-3}} by \frac{\left(xy\right)^{2}}{\left(z^{-1}\right)^{2}} by multiplying \frac{z^{-3}}{\left(xy^{5}\right)^{-3}} by the reciprocal of \frac{\left(xy\right)^{2}}{\left(z^{-1}\right)^{2}}.

$$\frac{z^{-3}\left(z^{-1}\right)^{2}}{\left(xy^{5}\right)^{-3}\left(xy\right)^{2}}$$

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

$$\frac{z^{-3}z^{-2}}{\left(xy^{5}\right)^{-3}\left(xy\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -3 and -2 to get -5.

$$\frac{z^{-5}}{\left(xy^{5}\right)^{-3}\left(xy\right)^{2}}$$

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

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

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

$$\frac{z^{-5}}{x^{-3}y^{-15}\left(xy\right)^{2}}$$

Expand \left(xy\right)^{2}.

$$\frac{z^{-5}}{x^{-3}y^{-15}x^{2}y^{2}}$$

To multiply powers of the same base, add their exponents. Add -3 and 2 to get -1.

$$\frac{z^{-5}}{x^{-1}y^{-15}y^{2}}$$

To multiply powers of the same base, add their exponents. Add -15 and 2 to get -13.

$$\frac{z^{-5}}{x^{-1}y^{-13}}$$

Expand

$$\frac{xy^{13}}{z^{5}}$$

Solution Steps

Cancel out x in both numerator and denominator.

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

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

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

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

$$\frac{\left(\frac{z^{1}}{xy^{5}}\right)^{-3}}{\left(\frac{xy}{z^{-1}}\right)^{2}}$$

Calculate z to the power of 1 and get z.

$$\frac{\left(\frac{z}{xy^{5}}\right)^{-3}}{\left(\frac{xy}{z^{-1}}\right)^{2}}$$

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

$$\frac{\frac{z^{-3}}{\left(xy^{5}\right)^{-3}}}{\left(\frac{xy}{z^{-1}}\right)^{2}}$$

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

$$\frac{\frac{z^{-3}}{\left(xy^{5}\right)^{-3}}}{\frac{\left(xy\right)^{2}}{\left(z^{-1}\right)^{2}}}$$

Divide \frac{z^{-3}}{\left(xy^{5}\right)^{-3}} by \frac{\left(xy\right)^{2}}{\left(z^{-1}\right)^{2}} by multiplying \frac{z^{-3}}{\left(xy^{5}\right)^{-3}} by the reciprocal of \frac{\left(xy\right)^{2}}{\left(z^{-1}\right)^{2}}.

$$\frac{z^{-3}\left(z^{-1}\right)^{2}}{\left(xy^{5}\right)^{-3}\left(xy\right)^{2}}$$

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

$$\frac{z^{-3}z^{-2}}{\left(xy^{5}\right)^{-3}\left(xy\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -3 and -2 to get -5.

$$\frac{z^{-5}}{\left(xy^{5}\right)^{-3}\left(xy\right)^{2}}$$

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

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

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

$$\frac{z^{-5}}{x^{-3}y^{-15}\left(xy\right)^{2}}$$

Expand \left(xy\right)^{2}.

$$\frac{z^{-5}}{x^{-3}y^{-15}x^{2}y^{2}}$$

To multiply powers of the same base, add their exponents. Add -3 and 2 to get -1.

$$\frac{z^{-5}}{x^{-1}y^{-15}y^{2}}$$

To multiply powers of the same base, add their exponents. Add -15 and 2 to get -13.

$$\frac{z^{-5}}{x^{-1}y^{-13}}$$