Solve frac{x^{2}y^{3}z}{x^{-3}y^{2}z^{-1}}:xy^-2z^-1/x^2y^-3z=

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

Evaluate

$$z^{4}x^{6}$$

Solution Steps

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

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

Cancel out xy^{2} in both numerator and denominator.

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

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

$$y^{-3}zx^{2}y^{3}z^{3}x^{4}$$

Multiply y^{-3} and y^{3} to get 1.

$$zx^{2}z^{3}x^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$z^{4}x^{2}x^{4}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$z^{4}x^{6}$$

Differentiate w.r.t. x

$$6z^{4}x^{5}$$