Solve (-2xy^{2}z^{-1})^2*(3x^2y^3z)^-1

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

Evaluate

$$\frac{4y}{3z^{3}}$$

Solution Steps

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

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

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

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

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

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

Calculate -2 to the power of 2 and get 4.

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

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

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

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

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

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

$$4x^{2}y^{4}z^{-2}\times 3^{-1}x^{-2}y^{-3}z^{-1}$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$4x^{2}y^{4}z^{-2}\times \frac{1}{3}x^{-2}y^{-3}z^{-1}$$

Multiply 4 and \frac{1}{3} to get \frac{4}{3}.

$$\frac{4}{3}x^{2}y^{4}z^{-2}x^{-2}y^{-3}z^{-1}$$

Multiply x^{2} and x^{-2} to get 1.

$$\frac{4}{3}y^{4}z^{-2}y^{-3}z^{-1}$$

To multiply powers of the same base, add their exponents. Add 4 and -3 to get 1.

$$\frac{4}{3}y^{1}z^{-2}z^{-1}$$

To multiply powers of the same base, add their exponents. Add -2 and -1 to get -3.

$$\frac{4}{3}y^{1}z^{-3}$$

Calculate y to the power of 1 and get y.

$$\frac{4}{3}yz^{-3}$$

Expand

$$\frac{4y}{3z^{3}}$$

Solution Steps

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

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

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

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

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

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

Calculate -2 to the power of 2 and get 4.

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

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

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

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

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

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

$$4x^{2}y^{4}z^{-2}\times 3^{-1}x^{-2}y^{-3}z^{-1}$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$4x^{2}y^{4}z^{-2}\times \frac{1}{3}x^{-2}y^{-3}z^{-1}$$

Multiply 4 and \frac{1}{3} to get \frac{4}{3}.

$$\frac{4}{3}x^{2}y^{4}z^{-2}x^{-2}y^{-3}z^{-1}$$

Multiply x^{2} and x^{-2} to get 1.

$$\frac{4}{3}y^{4}z^{-2}y^{-3}z^{-1}$$

To multiply powers of the same base, add their exponents. Add 4 and -3 to get 1.

$$\frac{4}{3}y^{1}z^{-2}z^{-1}$$

To multiply powers of the same base, add their exponents. Add -2 and -1 to get -3.

$$\frac{4}{3}y^{1}z^{-3}$$

Calculate y to the power of 1 and get y.

$$\frac{4}{3}yz^{-3}$$