Solve frac{(4x^{6}y^2)^frac{1{2}}}{x^-2z^-1}

$$\frac { ( 4 x ^ { 6 } y ^ { 2 } ) ^ { \frac { 1 } { 2 } } } { x ^ { - 2 } z ^ { - 1 } }$$

Evaluate

$$2z|y|x^{2}\left(|x|\right)^{3}$$

Solution Steps

Expand \left(4x^{6}y^{2}\right)^{\frac{1}{2}}.

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

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

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

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

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

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

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

Calculate y to the power of 1 and get y.

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

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

$$\frac{2yx^{5}}{\frac{1}{z}}$$

Divide 2yx^{5} by \frac{1}{z} by multiplying 2yx^{5} by the reciprocal of \frac{1}{z}.

$$2yx^{5}z$$

Differentiate w.r.t. x

$$10z|x||y|x^{3}$$