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

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

Evaluate

$$\frac{2}{x^{8}}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{4x^{-6}}{\left(8x^{6}\right)^{\frac{1}{3}}}$$

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

$$\frac{4x^{-6}}{8^{\frac{1}{3}}\left(x^{6}\right)^{\frac{1}{3}}}$$

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

$$\frac{4x^{-6}}{8^{\frac{1}{3}}x^{2}}$$

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

$$\frac{4x^{-6}}{2x^{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2x^{-6}}{x^{2}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{2}{x^{8}}$$

Differentiate w.r.t. x

$$-\frac{16}{x^{9}}$$