Solve (x^3y^-2z^21/x^-5y)div(frac{{x}^{4}{y}^-1{z}^10}{{x}^-7{y}^3})

$$( \frac{ { x }^{ 3 } { y }^{ -2 } { z }^{ 21 } }{ { x }^{ -5 } y } ) \div ( \frac{ { x }^{ 4 } { y }^{ -1 } { z }^{ 10 } }{ { x }^{ -7 } { y }^{ 3 } } )$$

Evaluate

$$\frac{yz^{11}}{x^{3}}$$

Solution Steps

Divide \frac{x^{3}y^{-2}z^{21}}{x^{-5}y} by \frac{x^{4}y^{-1}z^{10}}{x^{-7}y^{3}} by multiplying \frac{x^{3}y^{-2}z^{21}}{x^{-5}y} by the reciprocal of \frac{x^{4}y^{-1}z^{10}}{x^{-7}y^{3}}.

$$\frac{x^{3}y^{-2}z^{21}x^{-7}y^{3}}{x^{-5}yx^{4}y^{-1}z^{10}}$$

Cancel out yx^{3}z^{10} in both numerator and denominator.

$$\frac{x^{-7}y^{-2}y^{2}z^{11}}{x^{-5}\times \frac{1}{y}x}$$

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

$$\frac{x^{-7}y^{-2}y^{3}z^{11}}{x^{-5}x}$$

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

$$\frac{y^{-2}y^{3}z^{11}}{xx^{2}}$$

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$\frac{y^{1}z^{11}}{xx^{2}}$$

Calculate y to the power of 1 and get y.

$$\frac{yz^{11}}{xx^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{yz^{11}}{x^{3}}$$

Differentiate w.r.t. x

$$-\frac{3yz^{11}}{x^{4}}$$