Solve (frac{x^{-2}y^{-5}}{z^{-4}})div(xz^-2)^-4/(xy^-3)^3

$$( \frac { x ^ { - 2 } y ^ { - 5 } } { z ^ { - 4 } } ) \div \frac { ( x z ^ { - 2 } ) ^ { - 4 } } { ( x y ^ { - 3 } ) ^ { 3 } }$$

Evaluate

$$\frac{x^{5}}{z^{4}y^{14}}$$

Solution Steps

Divide \frac{x^{-2}y^{-5}}{z^{-4}} by \frac{\left(xz^{-2}\right)^{-4}}{\left(xy^{-3}\right)^{3}} by multiplying \frac{x^{-2}y^{-5}}{z^{-4}} by the reciprocal of \frac{\left(xz^{-2}\right)^{-4}}{\left(xy^{-3}\right)^{3}}.

$$\frac{x^{-2}y^{-5}\left(xy^{-3}\right)^{3}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

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

$$\frac{x^{-2}y^{-5}x^{3}\left(y^{-3}\right)^{3}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{x^{-2}y^{-5}x^{3}y^{-9}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

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

$$\frac{x^{1}y^{-5}y^{-9}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -5 and -9 to get -14.

$$\frac{x^{1}y^{-14}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

Expand \left(xz^{-2}\right)^{-4}.

$$\frac{x^{1}y^{-14}}{z^{-4}x^{-4}\left(z^{-2}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -4 to get 8.

$$\frac{x^{1}y^{-14}}{z^{-4}x^{-4}z^{8}}$$

To multiply powers of the same base, add their exponents. Add -4 and 8 to get 4.

$$\frac{x^{1}y^{-14}}{z^{4}x^{-4}}$$

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

$$\frac{y^{-14}x^{5}}{z^{4}}$$

Expand

$$\frac{x^{5}}{z^{4}y^{14}}$$

Solution Steps

Divide \frac{x^{-2}y^{-5}}{z^{-4}} by \frac{\left(xz^{-2}\right)^{-4}}{\left(xy^{-3}\right)^{3}} by multiplying \frac{x^{-2}y^{-5}}{z^{-4}} by the reciprocal of \frac{\left(xz^{-2}\right)^{-4}}{\left(xy^{-3}\right)^{3}}.

$$\frac{x^{-2}y^{-5}\left(xy^{-3}\right)^{3}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

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

$$\frac{x^{-2}y^{-5}x^{3}\left(y^{-3}\right)^{3}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{x^{-2}y^{-5}x^{3}y^{-9}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

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

$$\frac{x^{1}y^{-5}y^{-9}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -5 and -9 to get -14.

$$\frac{x^{1}y^{-14}}{z^{-4}\left(xz^{-2}\right)^{-4}}$$

Expand \left(xz^{-2}\right)^{-4}.

$$\frac{x^{1}y^{-14}}{z^{-4}x^{-4}\left(z^{-2}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -4 to get 8.

$$\frac{x^{1}y^{-14}}{z^{-4}x^{-4}z^{8}}$$

To multiply powers of the same base, add their exponents. Add -4 and 8 to get 4.

$$\frac{x^{1}y^{-14}}{z^{4}x^{-4}}$$

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

$$\frac{y^{-14}x^{5}}{z^{4}}$$