Solve frac{(z^{-6}y^{10})}{[(z^{4}z^{4})]^{6}}*frac{(z^{-5}y^{10})^{-4}}{[(z^2y^4)^6]^2}

$$\frac { ( z ^ { - 6 } y ^ { 10 } ) } { [ ( z ^ { 4 } z ^ { 4 } ) ] ^ { 6 } } \cdot \frac { ( z ^ { - 5 } y ^ { 10 } ) ^ { - 4 } } { [ ( z ^ { 2 } y ^ { 4 } ) ^ { 6 } ] ^ { 2 } }$$

Evaluate

$$\frac{1}{z^{58}y^{78}}$$

Solution Steps

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

$$\frac{z^{-6}y^{10}}{\left(z^{8}\right)^{6}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(\left(z^{2}y^{4}\right)^{6}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and 6 to get 48.

$$\frac{z^{-6}y^{10}}{z^{48}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(\left(z^{2}y^{4}\right)^{6}\right)^{2}}$$

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

$$\frac{z^{-6}y^{10}}{z^{48}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

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

$$\frac{y^{10}}{z^{54}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

Expand \left(z^{-5}y^{10}\right)^{-4}.

$$\frac{y^{10}}{z^{54}}\times \frac{\left(z^{-5}\right)^{-4}\left(y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

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

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}\left(y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

To raise a power to another power, multiply the exponents. Multiply 10 and -4 to get -40.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{\left(z^{2}y^{4}\right)^{12}}$$

Expand \left(z^{2}y^{4}\right)^{12}.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{\left(z^{2}\right)^{12}\left(y^{4}\right)^{12}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 12 to get 24.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{z^{24}\left(y^{4}\right)^{12}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 12 to get 48.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{z^{24}y^{48}}$$

Cancel out z^{20} in both numerator and denominator.

$$\frac{y^{10}}{z^{54}}\times \frac{y^{-40}}{z^{4}y^{48}}$$

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

$$\frac{y^{10}}{z^{54}}\times \frac{1}{z^{4}y^{88}}$$

Multiply \frac{y^{10}}{z^{54}} times \frac{1}{z^{4}y^{88}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{y^{10}}{z^{54}z^{4}y^{88}}$$

Cancel out y^{10} in both numerator and denominator.

$$\frac{1}{z^{4}z^{54}y^{78}}$$

To multiply powers of the same base, add their exponents. Add 4 and 54 to get 58.

$$\frac{1}{z^{58}y^{78}}$$

Expand

$$\frac{1}{z^{58}y^{78}}$$

Solution Steps

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

$$\frac{z^{-6}y^{10}}{\left(z^{8}\right)^{6}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(\left(z^{2}y^{4}\right)^{6}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and 6 to get 48.

$$\frac{z^{-6}y^{10}}{z^{48}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(\left(z^{2}y^{4}\right)^{6}\right)^{2}}$$

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

$$\frac{z^{-6}y^{10}}{z^{48}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

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

$$\frac{y^{10}}{z^{54}}\times \frac{\left(z^{-5}y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

Expand \left(z^{-5}y^{10}\right)^{-4}.

$$\frac{y^{10}}{z^{54}}\times \frac{\left(z^{-5}\right)^{-4}\left(y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

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

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}\left(y^{10}\right)^{-4}}{\left(z^{2}y^{4}\right)^{12}}$$

To raise a power to another power, multiply the exponents. Multiply 10 and -4 to get -40.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{\left(z^{2}y^{4}\right)^{12}}$$

Expand \left(z^{2}y^{4}\right)^{12}.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{\left(z^{2}\right)^{12}\left(y^{4}\right)^{12}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 12 to get 24.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{z^{24}\left(y^{4}\right)^{12}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 12 to get 48.

$$\frac{y^{10}}{z^{54}}\times \frac{z^{20}y^{-40}}{z^{24}y^{48}}$$

Cancel out z^{20} in both numerator and denominator.

$$\frac{y^{10}}{z^{54}}\times \frac{y^{-40}}{z^{4}y^{48}}$$

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

$$\frac{y^{10}}{z^{54}}\times \frac{1}{z^{4}y^{88}}$$

Multiply \frac{y^{10}}{z^{54}} times \frac{1}{z^{4}y^{88}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{y^{10}}{z^{54}z^{4}y^{88}}$$

Cancel out y^{10} in both numerator and denominator.

$$\frac{1}{z^{4}z^{54}y^{78}}$$

To multiply powers of the same base, add their exponents. Add 4 and 54 to get 58.

$$\frac{1}{z^{58}y^{78}}$$