Solve (8x^{-3}y^6)^-frac{1{3}}(2x^-3y^-1)

$$( 8 x ^ { - 3 } y ^ { 6 } ) ^ { - \frac { 1 } { 3 } } ( 2 x ^ { - 3 } y ^ { - 1 } )$$

Evaluate

$$\frac{1}{x^{2}y^{3}}$$

Solution Steps

Expand \left(8x^{-3}y^{6}\right)^{-\frac{1}{3}}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 6 and -\frac{1}{3} to get -2.

$$8^{-\frac{1}{3}}x^{1}y^{-2}\times 2x^{-3}y^{-1}$$

Calculate 8 to the power of -\frac{1}{3} and get \frac{1}{2}.

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

Calculate x to the power of 1 and get x.

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

Multiply \frac{1}{2} and 2 to get 1.

$$xy^{-2}x^{-3}y^{-1}$$

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

$$x^{-2}y^{-2}y^{-1}$$

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

$$x^{-2}y^{-3}$$

Differentiate w.r.t. x

$$-\frac{2}{\left(xy\right)^{3}}$$