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

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

Evaluate

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

Solution Steps

Expand \left(x^{-3}y^{-1}z^{4}\right)^{-2}.

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

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

$$\frac{x^{6}\left(y^{-1}\right)^{-2}\left(z^{4}\right)^{-2}}{\left(xy^{2}z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}\left(z^{4}\right)^{-2}}{\left(xy^{2}z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{\left(xy^{2}z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{x^{3}\left(y^{2}\right)^{3}\left(z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{x^{3}y^{6}\left(z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{x^{3}y^{6}z^{-6}}$$

Cancel out y^{2}x^{3} in both numerator and denominator.

$$\frac{z^{-8}x^{3}}{z^{-6}y^{4}}$$

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

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

Expand

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

Solution Steps

Expand \left(x^{-3}y^{-1}z^{4}\right)^{-2}.

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

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

$$\frac{x^{6}\left(y^{-1}\right)^{-2}\left(z^{4}\right)^{-2}}{\left(xy^{2}z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}\left(z^{4}\right)^{-2}}{\left(xy^{2}z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{\left(xy^{2}z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{x^{3}\left(y^{2}\right)^{3}\left(z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{x^{3}y^{6}\left(z^{-2}\right)^{3}}$$

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

$$\frac{x^{6}y^{2}z^{-8}}{x^{3}y^{6}z^{-6}}$$

Cancel out y^{2}x^{3} in both numerator and denominator.

$$\frac{z^{-8}x^{3}}{z^{-6}y^{4}}$$

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

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