Solve frac{(32y^{-5}z^{10})^1/5}{(64y^6z^-12)^-1/6}

$$\frac { ( 32 y ^ { - 5 } z ^ { 10 } ) ^ { 1 / 5 } } { ( 64 y ^ { 6 } z ^ { - 12 } ) ^ { - 1 / 6 } }$$

Evaluate

$$\frac{4|y|}{y}$$

Solution Steps

Expand \left(32y^{-5}z^{10}\right)^{\frac{1}{5}}.

$$\frac{32^{\frac{1}{5}}\left(y^{-5}\right)^{\frac{1}{5}}\left(z^{10}\right)^{\frac{1}{5}}}{\left(64y^{6}z^{-12}\right)^{-\frac{1}{6}}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and \frac{1}{5} to get -1.

$$\frac{32^{\frac{1}{5}}y^{-1}\left(z^{10}\right)^{\frac{1}{5}}}{\left(64y^{6}z^{-12}\right)^{-\frac{1}{6}}}$$

To raise a power to another power, multiply the exponents. Multiply 10 and \frac{1}{5} to get 2.

$$\frac{32^{\frac{1}{5}}y^{-1}z^{2}}{\left(64y^{6}z^{-12}\right)^{-\frac{1}{6}}}$$

Calculate 32 to the power of \frac{1}{5} and get 2.

$$\frac{2y^{-1}z^{2}}{\left(64y^{6}z^{-12}\right)^{-\frac{1}{6}}}$$

Expand \left(64y^{6}z^{-12}\right)^{-\frac{1}{6}}.

$$\frac{2y^{-1}z^{2}}{64^{-\frac{1}{6}}\left(y^{6}\right)^{-\frac{1}{6}}\left(z^{-12}\right)^{-\frac{1}{6}}}$$

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

$$\frac{2y^{-1}z^{2}}{64^{-\frac{1}{6}}y^{-1}\left(z^{-12}\right)^{-\frac{1}{6}}}$$

To raise a power to another power, multiply the exponents. Multiply -12 and -\frac{1}{6} to get 2.

$$\frac{2y^{-1}z^{2}}{64^{-\frac{1}{6}}y^{-1}z^{2}}$$

Calculate 64 to the power of -\frac{1}{6} and get \frac{1}{2}.

$$\frac{2y^{-1}z^{2}}{\frac{1}{2}y^{-1}z^{2}}$$

Cancel out \frac{1}{y}z^{2} in both numerator and denominator.

$$\frac{2}{\frac{1}{2}}$$

Divide 2 by \frac{1}{2} by multiplying 2 by the reciprocal of \frac{1}{2}.

$$2\times 2$$

Multiply 2 and 2 to get 4.

$$4$$

Differentiate w.r.t. y

$$0$$