Solve (x^-2z^4)^-1*(x/z)^-2

$$( x ^ { - 2 } z ^ { 4 } ) ^ { - 1 } \cdot ( \frac { x } { z } ) ^ { - 2 }$$

Evaluate

$$\frac{1}{z^{2}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

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

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

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

Use the Commutative Property of Multiplication.

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

To raise a power to another power, multiply the exponents.

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

Multiply -2 times -1.

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

Multiply 4 times -1.

$$x^{2}x^{-2}z^{-4}z^{-\left(-2\right)}$$

Multiply -1 times -2.

$$x^{2}x^{-2}z^{-4}z^{2}$$

To multiply powers of the same base, add their exponents.

$$x^{2-2}z^{-4+2}$$

Add the exponents 2 and -2.

$$x^{0}z^{-4+2}$$

Add the exponents -4 and 2.

$$x^{0}z^{-2}$$

For any term t except 0, t^{0}=1.

$$1z^{-2}$$

For any term t, t\times 1=t and 1t=t.

$$z^{-2}$$

Expand

$$\frac{1}{z^{2}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

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

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

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

Use the Commutative Property of Multiplication.

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

To raise a power to another power, multiply the exponents.

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

Multiply -2 times -1.

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

Multiply 4 times -1.

$$x^{2}x^{-2}z^{-4}z^{-\left(-2\right)}$$

Multiply -1 times -2.

$$x^{2}x^{-2}z^{-4}z^{2}$$

To multiply powers of the same base, add their exponents.

$$x^{2-2}z^{-4+2}$$

Add the exponents 2 and -2.

$$x^{0}z^{-4+2}$$

Add the exponents -4 and 2.

$$x^{0}z^{-2}$$

For any term t except 0, t^{0}=1.

$$1z^{-2}$$

For any term t, t\times 1=t and 1t=t.

$$z^{-2}$$