Solve (8x^{3}y^{-5}z^{4})^{-2}(2x^2y^-4z^3)^3

$$( 8 x ^ { 3 } y ^ { - 5 } z ^ { 4 } ) ^ { - 2 } ( 2 x ^ { 2 } y ^ { - 4 } z ^ { 3 } ) ^ { 3 }$$

Evaluate

$$\frac{z}{8y^{2}}$$

Solution Steps

Expand \left(8x^{3}y^{-5}z^{4}\right)^{-2}.

$$8^{-2}\left(x^{3}\right)^{-2}\left(y^{-5}\right)^{-2}\left(z^{4}\right)^{-2}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

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

$$8^{-2}x^{-6}\left(y^{-5}\right)^{-2}\left(z^{4}\right)^{-2}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -2 to get 10.

$$8^{-2}x^{-6}y^{10}\left(z^{4}\right)^{-2}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -2 to get -8.

$$8^{-2}x^{-6}y^{10}z^{-8}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

Expand \left(2x^{2}y^{-4}z^{3}\right)^{3}.

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}\left(x^{2}\right)^{3}\left(y^{-4}\right)^{3}\left(z^{3}\right)^{3}$$

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

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}x^{6}\left(y^{-4}\right)^{3}\left(z^{3}\right)^{3}$$

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

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}x^{6}y^{-12}\left(z^{3}\right)^{3}$$

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

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}x^{6}y^{-12}z^{9}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 8x^{6}y^{-12}z^{9}$$

Multiply \frac{1}{64} and 8 to get \frac{1}{8}.

$$\frac{1}{8}x^{-6}y^{10}z^{-8}x^{6}y^{-12}z^{9}$$

Multiply x^{-6} and x^{6} to get 1.

$$\frac{1}{8}y^{10}z^{-8}y^{-12}z^{9}$$

To multiply powers of the same base, add their exponents. Add 10 and -12 to get -2.

$$\frac{1}{8}y^{-2}z^{-8}z^{9}$$

To multiply powers of the same base, add their exponents. Add -8 and 9 to get 1.

$$\frac{1}{8}y^{-2}z^{1}$$

Calculate z to the power of 1 and get z.

$$\frac{1}{8}y^{-2}z$$

Expand

$$\frac{z}{8y^{2}}$$

Solution Steps

Expand \left(8x^{3}y^{-5}z^{4}\right)^{-2}.

$$8^{-2}\left(x^{3}\right)^{-2}\left(y^{-5}\right)^{-2}\left(z^{4}\right)^{-2}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

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

$$8^{-2}x^{-6}\left(y^{-5}\right)^{-2}\left(z^{4}\right)^{-2}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -2 to get 10.

$$8^{-2}x^{-6}y^{10}\left(z^{4}\right)^{-2}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -2 to get -8.

$$8^{-2}x^{-6}y^{10}z^{-8}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times \left(2x^{2}y^{-4}z^{3}\right)^{3}$$

Expand \left(2x^{2}y^{-4}z^{3}\right)^{3}.

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}\left(x^{2}\right)^{3}\left(y^{-4}\right)^{3}\left(z^{3}\right)^{3}$$

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

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}x^{6}\left(y^{-4}\right)^{3}\left(z^{3}\right)^{3}$$

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

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}x^{6}y^{-12}\left(z^{3}\right)^{3}$$

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

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 2^{3}x^{6}y^{-12}z^{9}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{1}{64}x^{-6}y^{10}z^{-8}\times 8x^{6}y^{-12}z^{9}$$

Multiply \frac{1}{64} and 8 to get \frac{1}{8}.

$$\frac{1}{8}x^{-6}y^{10}z^{-8}x^{6}y^{-12}z^{9}$$

Multiply x^{-6} and x^{6} to get 1.

$$\frac{1}{8}y^{10}z^{-8}y^{-12}z^{9}$$

To multiply powers of the same base, add their exponents. Add 10 and -12 to get -2.

$$\frac{1}{8}y^{-2}z^{-8}z^{9}$$

To multiply powers of the same base, add their exponents. Add -8 and 9 to get 1.

$$\frac{1}{8}y^{-2}z^{1}$$

Calculate z to the power of 1 and get z.

$$\frac{1}{8}y^{-2}z$$