Solve (8x^{2}y^{4})(4xy^-1/x^2y^3)^-1

$$( 8 x ^ { 2 } y ^ { 4 } ) ( \frac { 4 x y ^ { - 1 } } { x ^ { 2 } y ^ { 3 } } ) ^ { - 1 }$$

Evaluate

$$2x^{3}y^{8}$$

Solution Steps

Cancel out x in both numerator and denominator.

$$8x^{2}y^{4}\times \left(\frac{4\times \frac{1}{y}}{xy^{3}}\right)^{-1}$$

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

$$8x^{2}y^{4}\times \left(\frac{4}{xy^{4}}\right)^{-1}$$

To raise \frac{4}{xy^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$8x^{2}y^{4}\times \frac{4^{-1}}{\left(xy^{4}\right)^{-1}}$$

Express 8\times \frac{4^{-1}}{\left(xy^{4}\right)^{-1}} as a single fraction.

$$\frac{8\times 4^{-1}}{\left(xy^{4}\right)^{-1}}x^{2}y^{4}$$

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

$$\frac{8\times \frac{1}{4}}{\left(xy^{4}\right)^{-1}}x^{2}y^{4}$$

Multiply 8 and \frac{1}{4} to get 2.

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

Expand \left(xy^{4}\right)^{-1}.

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

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

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

Express \frac{2}{x^{-1}y^{-4}}x^{2} as a single fraction.

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

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

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

Express \frac{2x^{3}}{y^{-4}}y^{4} as a single fraction.

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

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

$$2x^{3}y^{8}$$

Differentiate w.r.t. x

$$6x^{2}y^{8}$$