Solve frac{(x^{3}y^{2}z^{6})^7}{(x^6y^2z)^6}

$$\frac { ( x ^ { 3 } y ^ { 2 } z ^ { 6 } ) ^ { 7 } } { ( x ^ { 6 } y ^ { 2 } z ) ^ { 6 } }$$

Evaluate

$$\frac{y^{2}z^{36}}{x^{15}}$$

Solution Steps

Expand \left(x^{3}y^{2}z^{6}\right)^{7}.

$$\frac{\left(x^{3}\right)^{7}\left(y^{2}\right)^{7}\left(z^{6}\right)^{7}}{\left(x^{6}y^{2}z\right)^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 7 to get 21.

$$\frac{x^{21}\left(y^{2}\right)^{7}\left(z^{6}\right)^{7}}{\left(x^{6}y^{2}z\right)^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 7 to get 14.

$$\frac{x^{21}y^{14}\left(z^{6}\right)^{7}}{\left(x^{6}y^{2}z\right)^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 7 to get 42.

$$\frac{x^{21}y^{14}z^{42}}{\left(x^{6}y^{2}z\right)^{6}}$$

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

$$\frac{x^{21}y^{14}z^{42}}{\left(x^{6}\right)^{6}\left(y^{2}\right)^{6}z^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 6 to get 36.

$$\frac{x^{21}y^{14}z^{42}}{x^{36}\left(y^{2}\right)^{6}z^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$\frac{x^{21}y^{14}z^{42}}{x^{36}y^{12}z^{6}}$$

Cancel out z^{6}y^{12}x^{21} in both numerator and denominator.

$$\frac{y^{2}z^{36}}{x^{15}}$$

Expand

$$\frac{y^{2}z^{36}}{x^{15}}$$

Solution Steps

Expand \left(x^{3}y^{2}z^{6}\right)^{7}.

$$\frac{\left(x^{3}\right)^{7}\left(y^{2}\right)^{7}\left(z^{6}\right)^{7}}{\left(x^{6}y^{2}z\right)^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 7 to get 21.

$$\frac{x^{21}\left(y^{2}\right)^{7}\left(z^{6}\right)^{7}}{\left(x^{6}y^{2}z\right)^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 7 to get 14.

$$\frac{x^{21}y^{14}\left(z^{6}\right)^{7}}{\left(x^{6}y^{2}z\right)^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 7 to get 42.

$$\frac{x^{21}y^{14}z^{42}}{\left(x^{6}y^{2}z\right)^{6}}$$

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

$$\frac{x^{21}y^{14}z^{42}}{\left(x^{6}\right)^{6}\left(y^{2}\right)^{6}z^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 6 to get 36.

$$\frac{x^{21}y^{14}z^{42}}{x^{36}\left(y^{2}\right)^{6}z^{6}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$\frac{x^{21}y^{14}z^{42}}{x^{36}y^{12}z^{6}}$$

Cancel out z^{6}y^{12}x^{21} in both numerator and denominator.

$$\frac{y^{2}z^{36}}{x^{15}}$$