Solve frac{(8xy^{3}z^{4})^{3}yz}{(4x^2y^4)^3z^6}

$$\frac { ( 8 x y ^ { 3 } z ^ { 4 } ) ^ { 3 } y z } { ( 4 x ^ { 2 } y ^ { 4 } ) ^ { 3 } z ^ { 6 } }$$

Evaluate

$$\frac{8z^{7}}{y^{2}x^{3}}$$

Solution Steps

Cancel out z in both numerator and denominator.

$$\frac{y\times \left(8xy^{3}z^{4}\right)^{3}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

Expand \left(8xy^{3}z^{4}\right)^{3}.

$$\frac{y\times 8^{3}x^{3}\left(y^{3}\right)^{3}\left(z^{4}\right)^{3}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

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

$$\frac{y\times 8^{3}x^{3}y^{9}\left(z^{4}\right)^{3}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{y\times 8^{3}x^{3}y^{9}z^{12}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

Calculate 8 to the power of 3 and get 512.

$$\frac{y\times 512x^{3}y^{9}z^{12}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 1 and 9 to get 10.

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

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

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 4^{3}\left(x^{2}\right)^{3}\left(y^{4}\right)^{3}}$$

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

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 4^{3}x^{6}\left(y^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 4^{3}x^{6}y^{12}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 64x^{6}y^{12}}$$

Cancel out 64x^{3}z^{5}y^{10} in both numerator and denominator.

$$\frac{8z^{7}}{y^{2}x^{3}}$$

Expand

$$\frac{8z^{7}}{y^{2}x^{3}}$$

Solution Steps

Cancel out z in both numerator and denominator.

$$\frac{y\times \left(8xy^{3}z^{4}\right)^{3}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

Expand \left(8xy^{3}z^{4}\right)^{3}.

$$\frac{y\times 8^{3}x^{3}\left(y^{3}\right)^{3}\left(z^{4}\right)^{3}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

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

$$\frac{y\times 8^{3}x^{3}y^{9}\left(z^{4}\right)^{3}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{y\times 8^{3}x^{3}y^{9}z^{12}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

Calculate 8 to the power of 3 and get 512.

$$\frac{y\times 512x^{3}y^{9}z^{12}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 1 and 9 to get 10.

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times \left(4x^{2}y^{4}\right)^{3}}$$

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

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 4^{3}\left(x^{2}\right)^{3}\left(y^{4}\right)^{3}}$$

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

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 4^{3}x^{6}\left(y^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 4^{3}x^{6}y^{12}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{y^{10}\times 512x^{3}z^{12}}{z^{5}\times 64x^{6}y^{12}}$$

Cancel out 64x^{3}z^{5}y^{10} in both numerator and denominator.

$$\frac{8z^{7}}{y^{2}x^{3}}$$