Solve frac{(-6x^{3}y^{2}z^{2})^3}{(-12x^2yz^5)^4}

$$\frac { ( - 6 x ^ { 3 } y ^ { 2 } z ^ { 2 } ) ^ { 3 } } { ( - 12 x ^ { 2 } y z ^ { 5 } ) ^ { 4 } }$$

Evaluate

$$-\frac{xy^{2}}{96z^{14}}$$

Solution Steps

Expand \left(-6x^{3}y^{2}z^{2}\right)^{3}.

$$\frac{\left(-6\right)^{3}\left(x^{3}\right)^{3}\left(y^{2}\right)^{3}\left(z^{2}\right)^{3}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

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

$$\frac{\left(-6\right)^{3}x^{9}\left(y^{2}\right)^{3}\left(z^{2}\right)^{3}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

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

$$\frac{\left(-6\right)^{3}x^{9}y^{6}\left(z^{2}\right)^{3}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

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

$$\frac{\left(-6\right)^{3}x^{9}y^{6}z^{6}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

Calculate -6 to the power of 3 and get -216.

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

Expand \left(-12x^{2}yz^{5}\right)^{4}.

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12\right)^{4}\left(x^{2}\right)^{4}y^{4}\left(z^{5}\right)^{4}}$$

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

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12\right)^{4}x^{8}y^{4}\left(z^{5}\right)^{4}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12\right)^{4}x^{8}y^{4}z^{20}}$$

Calculate -12 to the power of 4 and get 20736.

$$\frac{-216x^{9}y^{6}z^{6}}{20736x^{8}y^{4}z^{20}}$$

Cancel out 216y^{4}z^{6}x^{8} in both numerator and denominator.

$$\frac{-xy^{2}}{96z^{14}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{xy^{2}}{-96z^{14}}$$

Expand

$$-\frac{xy^{2}}{96z^{14}}$$

Solution Steps

Expand \left(-6x^{3}y^{2}z^{2}\right)^{3}.

$$\frac{\left(-6\right)^{3}\left(x^{3}\right)^{3}\left(y^{2}\right)^{3}\left(z^{2}\right)^{3}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

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

$$\frac{\left(-6\right)^{3}x^{9}\left(y^{2}\right)^{3}\left(z^{2}\right)^{3}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

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

$$\frac{\left(-6\right)^{3}x^{9}y^{6}\left(z^{2}\right)^{3}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

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

$$\frac{\left(-6\right)^{3}x^{9}y^{6}z^{6}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

Calculate -6 to the power of 3 and get -216.

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12x^{2}yz^{5}\right)^{4}}$$

Expand \left(-12x^{2}yz^{5}\right)^{4}.

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12\right)^{4}\left(x^{2}\right)^{4}y^{4}\left(z^{5}\right)^{4}}$$

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

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12\right)^{4}x^{8}y^{4}\left(z^{5}\right)^{4}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$\frac{-216x^{9}y^{6}z^{6}}{\left(-12\right)^{4}x^{8}y^{4}z^{20}}$$

Calculate -12 to the power of 4 and get 20736.

$$\frac{-216x^{9}y^{6}z^{6}}{20736x^{8}y^{4}z^{20}}$$

Cancel out 216y^{4}z^{6}x^{8} in both numerator and denominator.

$$\frac{-xy^{2}}{96z^{14}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{xy^{2}}{-96z^{14}}$$