Solve frac{(2y^{-1}z^{2})^{2}(3y^{-2}z^{-3})^3}{(y^3z^2)^-1}

$$\frac { ( 2 y ^ { - 1 } z ^ { 2 } ) ^ { 2 } ( 3 y ^ { - 2 } z ^ { - 3 } ) ^ { 3 } } { ( y ^ { 3 } z ^ { 2 } ) ^ { - 1 } }$$

Evaluate

$$\frac{108}{z^{3}y^{5}}$$

Solution Steps

Expand \left(2y^{-1}z^{2}\right)^{2}.

$$\frac{2^{2}\left(y^{-1}\right)^{2}\left(z^{2}\right)^{2}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{2^{2}y^{-2}\left(z^{2}\right)^{2}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{2^{2}y^{-2}z^{4}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4y^{-2}z^{4}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{4y^{-2}z^{4}\times 3^{3}\left(y^{-2}\right)^{3}\left(z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{4y^{-2}z^{4}\times 3^{3}y^{-6}\left(z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{4y^{-2}z^{4}\times 3^{3}y^{-6}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{4y^{-2}z^{4}\times 27y^{-6}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

Multiply 4 and 27 to get 108.

$$\frac{108y^{-2}z^{4}y^{-6}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

To multiply powers of the same base, add their exponents. Add -2 and -6 to get -8.

$$\frac{108y^{-8}z^{4}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

To multiply powers of the same base, add their exponents. Add 4 and -9 to get -5.

$$\frac{108y^{-8}z^{-5}}{\left(y^{3}z^{2}\right)^{-1}}$$

Expand \left(y^{3}z^{2}\right)^{-1}.

$$\frac{108y^{-8}z^{-5}}{\left(y^{3}\right)^{-1}\left(z^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$\frac{108y^{-8}z^{-5}}{y^{-3}\left(z^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{108y^{-8}z^{-5}}{y^{-3}z^{-2}}$$

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

$$\frac{108}{z^{3}y^{5}}$$

Expand

$$\frac{108}{z^{3}y^{5}}$$

Solution Steps

Expand \left(2y^{-1}z^{2}\right)^{2}.

$$\frac{2^{2}\left(y^{-1}\right)^{2}\left(z^{2}\right)^{2}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 2 to get -2.

$$\frac{2^{2}y^{-2}\left(z^{2}\right)^{2}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{2^{2}y^{-2}z^{4}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4y^{-2}z^{4}\times \left(3y^{-2}z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{4y^{-2}z^{4}\times 3^{3}\left(y^{-2}\right)^{3}\left(z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{4y^{-2}z^{4}\times 3^{3}y^{-6}\left(z^{-3}\right)^{3}}{\left(y^{3}z^{2}\right)^{-1}}$$

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

$$\frac{4y^{-2}z^{4}\times 3^{3}y^{-6}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{4y^{-2}z^{4}\times 27y^{-6}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

Multiply 4 and 27 to get 108.

$$\frac{108y^{-2}z^{4}y^{-6}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

To multiply powers of the same base, add their exponents. Add -2 and -6 to get -8.

$$\frac{108y^{-8}z^{4}z^{-9}}{\left(y^{3}z^{2}\right)^{-1}}$$

To multiply powers of the same base, add their exponents. Add 4 and -9 to get -5.

$$\frac{108y^{-8}z^{-5}}{\left(y^{3}z^{2}\right)^{-1}}$$

Expand \left(y^{3}z^{2}\right)^{-1}.

$$\frac{108y^{-8}z^{-5}}{\left(y^{3}\right)^{-1}\left(z^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$\frac{108y^{-8}z^{-5}}{y^{-3}\left(z^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{108y^{-8}z^{-5}}{y^{-3}z^{-2}}$$

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

$$\frac{108}{z^{3}y^{5}}$$