Solve (64x^7y^3/27x^1y)^-frac{1{3}}

$$( \frac { 64 x ^ { 7 } y ^ { 3 } } { 27 x ^ { 1 } y } ) ^ { - \frac { 1 } { 3 } }$$

Evaluate

$$\frac{3}{4y^{\frac{2}{3}}x^{2}}$$

Solution Steps

Cancel out xy in both numerator and denominator.

$$\left(\frac{64y^{2}x^{6}}{27}\right)^{-\frac{1}{3}}$$

To raise \frac{64y^{2}x^{6}}{27} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(64y^{2}x^{6}\right)^{-\frac{1}{3}}}{27^{-\frac{1}{3}}}$$

Expand \left(64y^{2}x^{6}\right)^{-\frac{1}{3}}.

$$\frac{64^{-\frac{1}{3}}\left(y^{2}\right)^{-\frac{1}{3}}\left(x^{6}\right)^{-\frac{1}{3}}}{27^{-\frac{1}{3}}}$$

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

$$\frac{64^{-\frac{1}{3}}y^{-\frac{2}{3}}\left(x^{6}\right)^{-\frac{1}{3}}}{27^{-\frac{1}{3}}}$$

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

$$\frac{64^{-\frac{1}{3}}y^{-\frac{2}{3}}x^{-2}}{27^{-\frac{1}{3}}}$$

Calculate 64 to the power of -\frac{1}{3} and get \frac{1}{4}.

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

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

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

Divide \frac{1}{4}y^{-\frac{2}{3}}x^{-2} by \frac{1}{3} by multiplying \frac{1}{4}y^{-\frac{2}{3}}x^{-2} by the reciprocal of \frac{1}{3}.

$$\frac{1}{4}y^{-\frac{2}{3}}x^{-2}\times 3$$

Multiply \frac{1}{4} and 3 to get \frac{3}{4}.

$$\frac{3}{4}y^{-\frac{2}{3}}x^{-2}$$

Differentiate w.r.t. x

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