Solve (3x^{3}y^{-2}z^{-1})^-2(5xy^-2z^-1)^2

$$( 3 x ^ { 3 } y ^ { - 2 } z ^ { - 1 } ) ^ { - 2 } ( 5 x y ^ { - 2 } z ^ { - 1 } ) ^ { 2 }$$

Evaluate

$$\frac{25}{9x^{4}}$$

Solution Steps

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

$$3^{-2}\left(x^{3}\right)^{-2}\left(y^{-2}\right)^{-2}\left(z^{-1}\right)^{-2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

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

$$3^{-2}x^{-6}\left(y^{-2}\right)^{-2}\left(z^{-1}\right)^{-2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

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

$$3^{-2}x^{-6}y^{4}\left(z^{-1}\right)^{-2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

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

$$3^{-2}x^{-6}y^{4}z^{2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

Expand \left(5xy^{-2}z^{-1}\right)^{2}.

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 5^{2}x^{2}\left(y^{-2}\right)^{2}\left(z^{-1}\right)^{2}$$

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

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 5^{2}x^{2}y^{-4}\left(z^{-1}\right)^{2}$$

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

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 5^{2}x^{2}y^{-4}z^{-2}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 25x^{2}y^{-4}z^{-2}$$

Multiply \frac{1}{9} and 25 to get \frac{25}{9}.

$$\frac{25}{9}x^{-6}y^{4}z^{2}x^{2}y^{-4}z^{-2}$$

To multiply powers of the same base, add their exponents. Add -6 and 2 to get -4.

$$\frac{25}{9}x^{-4}y^{4}z^{2}y^{-4}z^{-2}$$

Multiply y^{4} and y^{-4} to get 1.

$$\frac{25}{9}x^{-4}z^{2}z^{-2}$$

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

$$\frac{25}{9}x^{-4}$$

Expand

$$\frac{25}{9x^{4}}$$

Solution Steps

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

$$3^{-2}\left(x^{3}\right)^{-2}\left(y^{-2}\right)^{-2}\left(z^{-1}\right)^{-2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

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

$$3^{-2}x^{-6}\left(y^{-2}\right)^{-2}\left(z^{-1}\right)^{-2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

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

$$3^{-2}x^{-6}y^{4}\left(z^{-1}\right)^{-2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

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

$$3^{-2}x^{-6}y^{4}z^{2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times \left(5xy^{-2}z^{-1}\right)^{2}$$

Expand \left(5xy^{-2}z^{-1}\right)^{2}.

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 5^{2}x^{2}\left(y^{-2}\right)^{2}\left(z^{-1}\right)^{2}$$

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

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 5^{2}x^{2}y^{-4}\left(z^{-1}\right)^{2}$$

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

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 5^{2}x^{2}y^{-4}z^{-2}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{1}{9}x^{-6}y^{4}z^{2}\times 25x^{2}y^{-4}z^{-2}$$

Multiply \frac{1}{9} and 25 to get \frac{25}{9}.

$$\frac{25}{9}x^{-6}y^{4}z^{2}x^{2}y^{-4}z^{-2}$$

To multiply powers of the same base, add their exponents. Add -6 and 2 to get -4.

$$\frac{25}{9}x^{-4}y^{4}z^{2}y^{-4}z^{-2}$$

Multiply y^{4} and y^{-4} to get 1.

$$\frac{25}{9}x^{-4}z^{2}z^{-2}$$

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

$$\frac{25}{9}x^{-4}$$