Solve (8*x^3*y^-4/3/27x^-6*y)^-2/3

$$( \frac { 8 \cdot x ^ { 3 } \cdot y ^ { - 4 / 3 } } { 27 x ^ { - 6 } \cdot y } ) ^ { - 2 / 3 }$$

Evaluate

$$\frac{9y^{\frac{14}{9}}}{4x^{6}}$$

Solution Steps

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

$$\left(\frac{8y^{-\frac{4}{3}}x^{9}}{27y}\right)^{-\frac{2}{3}}$$

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

$$\left(\frac{8x^{9}}{27y^{\frac{7}{3}}}\right)^{-\frac{2}{3}}$$

To raise \frac{8x^{9}}{27y^{\frac{7}{3}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8x^{9}\right)^{-\frac{2}{3}}}{\left(27y^{\frac{7}{3}}\right)^{-\frac{2}{3}}}$$

Expand \left(8x^{9}\right)^{-\frac{2}{3}}.

$$\frac{8^{-\frac{2}{3}}\left(x^{9}\right)^{-\frac{2}{3}}}{\left(27y^{\frac{7}{3}}\right)^{-\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 9 and -\frac{2}{3} to get -6.

$$\frac{8^{-\frac{2}{3}}x^{-6}}{\left(27y^{\frac{7}{3}}\right)^{-\frac{2}{3}}}$$

Calculate 8 to the power of -\frac{2}{3} and get \frac{1}{4}.

$$\frac{\frac{1}{4}x^{-6}}{\left(27y^{\frac{7}{3}}\right)^{-\frac{2}{3}}}$$

Expand \left(27y^{\frac{7}{3}}\right)^{-\frac{2}{3}}.

$$\frac{\frac{1}{4}x^{-6}}{27^{-\frac{2}{3}}\left(y^{\frac{7}{3}}\right)^{-\frac{2}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{7}{3} and -\frac{2}{3} to get -\frac{14}{9}.

$$\frac{\frac{1}{4}x^{-6}}{27^{-\frac{2}{3}}y^{-\frac{14}{9}}}$$

Calculate 27 to the power of -\frac{2}{3} and get \frac{1}{9}.

$$\frac{\frac{1}{4}x^{-6}}{\frac{1}{9}y^{-\frac{14}{9}}}$$

Differentiate w.r.t. x

$$-\frac{27y^{\frac{14}{9}}}{2x^{7}}$$