Solve (3y^{-3}z^{6})^2(y^4z^-5)^-3

$$( 3 y ^ { - 3 } z ^ { 6 } ) ^ { 2 } ( y ^ { 4 } z ^ { - 5 } ) ^ { - 3 }$$

Evaluate

$$\frac{9z^{27}}{y^{18}}$$

Solution Steps

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

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

Expand \left(y^{4}z^{-5}\right)^{-3}.

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

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

$$9y^{-6}z^{12}y^{-12}\left(z^{-5}\right)^{-3}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -3 to get 15.

$$9y^{-6}z^{12}y^{-12}z^{15}$$

To multiply powers of the same base, add their exponents. Add -6 and -12 to get -18.

$$9y^{-18}z^{12}z^{15}$$

To multiply powers of the same base, add their exponents. Add 12 and 15 to get 27.

$$9y^{-18}z^{27}$$

Expand

$$\frac{9z^{27}}{y^{18}}$$

Solution Steps

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

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

Expand \left(y^{4}z^{-5}\right)^{-3}.

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

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

$$9y^{-6}z^{12}y^{-12}\left(z^{-5}\right)^{-3}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -3 to get 15.

$$9y^{-6}z^{12}y^{-12}z^{15}$$

To multiply powers of the same base, add their exponents. Add -6 and -12 to get -18.

$$9y^{-18}z^{12}z^{15}$$

To multiply powers of the same base, add their exponents. Add 12 and 15 to get 27.

$$9y^{-18}z^{27}$$