Solve frac{(x^{2}y^{3}z^{-2})^{-3}(x^{3}yz^3)^-frac{1{2}}}{(xyz^-5)^-frac{5{2}}}

$$\frac { ( x ^ { 2 } y ^ { 3 } z ^ { - 2 } ) ^ { - 3 } ( x ^ { 3 } y z ^ { 3 } ) ^ { - \frac { 1 } { 2 } } } { ( x y z ^ { - 5 } ) ^ { - \frac { 5 } { 2 } } }$$

Evaluate

$$\frac{\left(\frac{xy}{z^{\frac{13}{5}}}\right)^{\frac{5}{2}}}{y^{9}\sqrt{yz^{3}x^{15}}}$$

Solution Steps

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

$$\frac{\left(x^{2}\right)^{-3}\left(y^{3}\right)^{-3}\left(z^{-2}\right)^{-3}\left(x^{3}yz^{3}\right)^{-\frac{1}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

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

$$\frac{x^{-6}\left(y^{3}\right)^{-3}\left(z^{-2}\right)^{-3}\left(x^{3}yz^{3}\right)^{-\frac{1}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

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

$$\frac{x^{-6}y^{-9}\left(z^{-2}\right)^{-3}\left(x^{3}yz^{3}\right)^{-\frac{1}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

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

$$\frac{x^{-6}y^{-9}z^{6}\left(x^{3}yz^{3}\right)^{-\frac{1}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

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

$$\frac{x^{-6}y^{-9}z^{6}\left(x^{3}\right)^{-\frac{1}{2}}y^{-\frac{1}{2}}\left(z^{3}\right)^{-\frac{1}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

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

$$\frac{x^{-6}y^{-9}z^{6}x^{-\frac{3}{2}}y^{-\frac{1}{2}}\left(z^{3}\right)^{-\frac{1}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

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

$$\frac{x^{-6}y^{-9}z^{6}x^{-\frac{3}{2}}y^{-\frac{1}{2}}z^{-\frac{3}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

To multiply powers of the same base, add their exponents. Add -6 and -\frac{3}{2} to get -\frac{15}{2}.

$$\frac{x^{-\frac{15}{2}}y^{-9}z^{6}y^{-\frac{1}{2}}z^{-\frac{3}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

To multiply powers of the same base, add their exponents. Add -9 and -\frac{1}{2} to get -\frac{19}{2}.

$$\frac{x^{-\frac{15}{2}}y^{-\frac{19}{2}}z^{6}z^{-\frac{3}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

To multiply powers of the same base, add their exponents. Add 6 and -\frac{3}{2} to get \frac{9}{2}.

$$\frac{x^{-\frac{15}{2}}y^{-\frac{19}{2}}z^{\frac{9}{2}}}{\left(xyz^{-5}\right)^{-\frac{5}{2}}}$$

Expand \left(xyz^{-5}\right)^{-\frac{5}{2}}.

$$\frac{x^{-\frac{15}{2}}y^{-\frac{19}{2}}z^{\frac{9}{2}}}{x^{-\frac{5}{2}}y^{-\frac{5}{2}}\left(z^{-5}\right)^{-\frac{5}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -\frac{5}{2} to get \frac{25}{2}.

$$\frac{x^{-\frac{15}{2}}y^{-\frac{19}{2}}z^{\frac{9}{2}}}{x^{-\frac{5}{2}}y^{-\frac{5}{2}}z^{\frac{25}{2}}}$$

Cancel out z^{\frac{9}{2}} in both numerator and denominator.

$$\frac{y^{-\frac{19}{2}}x^{-\frac{15}{2}}}{x^{-\frac{5}{2}}y^{-\frac{5}{2}}z^{8}}$$

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

$$\frac{1}{x^{5}y^{7}z^{8}}$$

Differentiate w.r.t. x

$$\frac{5\times \left(\frac{xy}{yz^{\frac{13}{5}}z^{3}x^{\frac{17}{3}}}\right)^{\frac{3}{2}}\left(\left(yz^{3}x^{\frac{17}{3}}\right)^{\frac{3}{2}}-3xyz^{3}\sqrt{yz^{3}x^{15}}\right)}{2z^{\frac{13}{5}}y^{8}\sqrt{yz^{3}x^{15}}}$$