Solve (9x^4y^6z^4/3xy^-6{z^{-2}})divfrac{{left(2x{y}^{2}{z}^{-3}right)}^2}{{left(9{x}^-1{y}^2right)}^-1}

$$( \frac{ 9 { x }^{ 4 } { y }^{ 6 } { z }^{ 4 } }{ 3x { y }^{ -6 } { z }^{ -2 } } ) \div \frac{ { \left(2x { y }^{ 2 } { z }^{ -3 } \right) }^{ 2 } }{ { \left(9 { x }^{ -1 } { y }^{ 2 } \right) }^{ -1 } }$$

Evaluate

$$\frac{x^{2}y^{6}z^{12}}{12}$$

Solution Steps

Divide \frac{9x^{4}y^{6}z^{4}}{3xy^{-6}z^{-2}} by \frac{\left(2xy^{2}z^{-3}\right)^{2}}{\left(9x^{-1}y^{2}\right)^{-1}} by multiplying \frac{9x^{4}y^{6}z^{4}}{3xy^{-6}z^{-2}} by the reciprocal of \frac{\left(2xy^{2}z^{-3}\right)^{2}}{\left(9x^{-1}y^{2}\right)^{-1}}.

$$\frac{9x^{4}y^{6}z^{4}\times \left(9x^{-1}y^{2}\right)^{-1}}{3xy^{-6}z^{-2}\times \left(2xy^{2}z^{-3}\right)^{2}}$$

Cancel out 3x in both numerator and denominator.

$$\frac{3\times \frac{1}{9\times \frac{1}{x}y^{2}}x^{3}z^{4}y^{6}}{\left(2z^{-3}xy^{2}\right)^{2}y^{-6}z^{-2}}$$

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

$$\frac{3\times \frac{1}{9\times \frac{1}{x}y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express 9\times \frac{1}{x} as a single fraction.

$$\frac{3\times \frac{1}{\frac{9}{x}y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express \frac{9}{x}y^{2} as a single fraction.

$$\frac{3\times \frac{1}{\frac{9y^{2}}{x}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Divide 1 by \frac{9y^{2}}{x} by multiplying 1 by the reciprocal of \frac{9y^{2}}{x}.

$$\frac{3\times \frac{x}{9y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express 3\times \frac{x}{9y^{2}} as a single fraction.

$$\frac{\frac{3x}{9y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\frac{x}{3y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

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

$$\frac{\frac{xx^{3}}{3y^{2}}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express \frac{xx^{3}}{3y^{2}}z^{6} as a single fraction.

$$\frac{\frac{xx^{3}z^{6}}{3y^{2}}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express \frac{xx^{3}z^{6}}{3y^{2}}y^{12} as a single fraction.

$$\frac{\frac{xx^{3}z^{6}y^{12}}{3y^{2}}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

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

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Expand \left(2z^{-3}xy^{2}\right)^{2}.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{2^{2}\left(z^{-3}\right)^{2}x^{2}\left(y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{2^{2}z^{-6}x^{2}\left(y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{2^{2}z^{-6}x^{2}y^{4}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{4z^{-6}x^{2}y^{4}}$$

Express \frac{\frac{xx^{3}z^{6}y^{10}}{3}}{4z^{-6}x^{2}y^{4}} as a single fraction.

$$\frac{xx^{3}z^{6}y^{10}}{3\times 4z^{-6}x^{2}y^{4}}$$

Cancel out xxy^{4} in both numerator and denominator.

$$\frac{x^{2}y^{6}z^{6}}{3\times 4z^{-6}}$$

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

$$\frac{x^{2}y^{6}z^{12}}{3\times 4}$$

Multiply 3 and 4 to get 12.

$$\frac{x^{2}y^{6}z^{12}}{12}$$

Expand

$$\frac{x^{2}y^{6}z^{12}}{12}$$

Solution Steps

Divide \frac{9x^{4}y^{6}z^{4}}{3xy^{-6}z^{-2}} by \frac{\left(2xy^{2}z^{-3}\right)^{2}}{\left(9x^{-1}y^{2}\right)^{-1}} by multiplying \frac{9x^{4}y^{6}z^{4}}{3xy^{-6}z^{-2}} by the reciprocal of \frac{\left(2xy^{2}z^{-3}\right)^{2}}{\left(9x^{-1}y^{2}\right)^{-1}}.

$$\frac{9x^{4}y^{6}z^{4}\times \left(9x^{-1}y^{2}\right)^{-1}}{3xy^{-6}z^{-2}\times \left(2xy^{2}z^{-3}\right)^{2}}$$

Cancel out 3x in both numerator and denominator.

$$\frac{3\times \frac{1}{9\times \frac{1}{x}y^{2}}x^{3}z^{4}y^{6}}{\left(2z^{-3}xy^{2}\right)^{2}y^{-6}z^{-2}}$$

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

$$\frac{3\times \frac{1}{9\times \frac{1}{x}y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express 9\times \frac{1}{x} as a single fraction.

$$\frac{3\times \frac{1}{\frac{9}{x}y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express \frac{9}{x}y^{2} as a single fraction.

$$\frac{3\times \frac{1}{\frac{9y^{2}}{x}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Divide 1 by \frac{9y^{2}}{x} by multiplying 1 by the reciprocal of \frac{9y^{2}}{x}.

$$\frac{3\times \frac{x}{9y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express 3\times \frac{x}{9y^{2}} as a single fraction.

$$\frac{\frac{3x}{9y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\frac{x}{3y^{2}}x^{3}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

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

$$\frac{\frac{xx^{3}}{3y^{2}}z^{6}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express \frac{xx^{3}}{3y^{2}}z^{6} as a single fraction.

$$\frac{\frac{xx^{3}z^{6}}{3y^{2}}y^{12}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Express \frac{xx^{3}z^{6}}{3y^{2}}y^{12} as a single fraction.

$$\frac{\frac{xx^{3}z^{6}y^{12}}{3y^{2}}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

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

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{\left(2z^{-3}xy^{2}\right)^{2}}$$

Expand \left(2z^{-3}xy^{2}\right)^{2}.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{2^{2}\left(z^{-3}\right)^{2}x^{2}\left(y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{2^{2}z^{-6}x^{2}\left(y^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{2^{2}z^{-6}x^{2}y^{4}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{xx^{3}z^{6}y^{10}}{3}}{4z^{-6}x^{2}y^{4}}$$

Express \frac{\frac{xx^{3}z^{6}y^{10}}{3}}{4z^{-6}x^{2}y^{4}} as a single fraction.

$$\frac{xx^{3}z^{6}y^{10}}{3\times 4z^{-6}x^{2}y^{4}}$$

Cancel out xxy^{4} in both numerator and denominator.

$$\frac{x^{2}y^{6}z^{6}}{3\times 4z^{-6}}$$

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

$$\frac{x^{2}y^{6}z^{12}}{3\times 4}$$

Multiply 3 and 4 to get 12.

$$\frac{x^{2}y^{6}z^{12}}{12}$$