Solve frac{3(x^{2}y^{-7})^{2}}{2(xy^{2})^{2}}div(xy)^-3/(3x^-2y^4)^2

$$\frac { 3 ( x ^ { 2 } y ^ { - 7 } ) ^ { 2 } } { 2 ( x y ^ { 2 } ) ^ { 2 } } \div \frac { ( x y ) ^ { - 3 } } { ( 3 x ^ { - 2 } y ^ { 4 } ) ^ { 2 } }$$

Evaluate

$$\frac{27x}{2y^{7}}$$

Solution Steps

Divide \frac{3\left(x^{2}y^{-7}\right)^{2}}{2\left(xy^{2}\right)^{2}} by \frac{\left(xy\right)^{-3}}{\left(3x^{-2}y^{4}\right)^{2}} by multiplying \frac{3\left(x^{2}y^{-7}\right)^{2}}{2\left(xy^{2}\right)^{2}} by the reciprocal of \frac{\left(xy\right)^{-3}}{\left(3x^{-2}y^{4}\right)^{2}}.

$$\frac{3\left(x^{2}y^{-7}\right)^{2}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Expand \left(x^{2}y^{-7}\right)^{2}.

$$\frac{3\left(x^{2}\right)^{2}\left(y^{-7}\right)^{2}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{3x^{4}\left(y^{-7}\right)^{2}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply -7 and 2 to get -14.

$$\frac{3x^{4}y^{-14}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Expand \left(3x^{-2}y^{4}\right)^{2}.

$$\frac{3x^{4}y^{-14}\times 3^{2}\left(x^{-2}\right)^{2}\left(y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{3x^{4}y^{-14}\times 3^{2}x^{-4}\left(y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{3x^{4}y^{-14}\times 3^{2}x^{-4}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3x^{4}y^{-14}\times 9x^{-4}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Multiply 3 and 9 to get 27.

$$\frac{27x^{4}y^{-14}x^{-4}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Multiply x^{4} and x^{-4} to get 1.

$$\frac{27y^{-14}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

To multiply powers of the same base, add their exponents. Add -14 and 8 to get -6.

$$\frac{27y^{-6}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{2}\left(y^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{2}y^{4}\left(xy\right)^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{2}y^{4}x^{-3}y^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{-1}y^{4}y^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{-1}y^{1}}$$

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

$$\frac{27}{2\times \frac{1}{x}y^{7}}$$

Express 2\times \frac{1}{x} as a single fraction.

$$\frac{27}{\frac{2}{x}y^{7}}$$

Express \frac{2}{x}y^{7} as a single fraction.

$$\frac{27}{\frac{2y^{7}}{x}}$$

Divide 27 by \frac{2y^{7}}{x} by multiplying 27 by the reciprocal of \frac{2y^{7}}{x}.

$$\frac{27x}{2y^{7}}$$

Expand

$$\frac{27x}{2y^{7}}$$

Solution Steps

Divide \frac{3\left(x^{2}y^{-7}\right)^{2}}{2\left(xy^{2}\right)^{2}} by \frac{\left(xy\right)^{-3}}{\left(3x^{-2}y^{4}\right)^{2}} by multiplying \frac{3\left(x^{2}y^{-7}\right)^{2}}{2\left(xy^{2}\right)^{2}} by the reciprocal of \frac{\left(xy\right)^{-3}}{\left(3x^{-2}y^{4}\right)^{2}}.

$$\frac{3\left(x^{2}y^{-7}\right)^{2}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Expand \left(x^{2}y^{-7}\right)^{2}.

$$\frac{3\left(x^{2}\right)^{2}\left(y^{-7}\right)^{2}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{3x^{4}\left(y^{-7}\right)^{2}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply -7 and 2 to get -14.

$$\frac{3x^{4}y^{-14}\times \left(3x^{-2}y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Expand \left(3x^{-2}y^{4}\right)^{2}.

$$\frac{3x^{4}y^{-14}\times 3^{2}\left(x^{-2}\right)^{2}\left(y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{3x^{4}y^{-14}\times 3^{2}x^{-4}\left(y^{4}\right)^{2}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{3x^{4}y^{-14}\times 3^{2}x^{-4}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3x^{4}y^{-14}\times 9x^{-4}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Multiply 3 and 9 to get 27.

$$\frac{27x^{4}y^{-14}x^{-4}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

Multiply x^{4} and x^{-4} to get 1.

$$\frac{27y^{-14}y^{8}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

To multiply powers of the same base, add their exponents. Add -14 and 8 to get -6.

$$\frac{27y^{-6}}{2\left(xy^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{2}\left(y^{2}\right)^{2}\left(xy\right)^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{2}y^{4}\left(xy\right)^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{2}y^{4}x^{-3}y^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{-1}y^{4}y^{-3}}$$

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

$$\frac{27y^{-6}}{2x^{-1}y^{1}}$$

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

$$\frac{27}{2\times \frac{1}{x}y^{7}}$$

Express 2\times \frac{1}{x} as a single fraction.

$$\frac{27}{\frac{2}{x}y^{7}}$$

Express \frac{2}{x}y^{7} as a single fraction.

$$\frac{27}{\frac{2y^{7}}{x}}$$

Divide 27 by \frac{2y^{7}}{x} by multiplying 27 by the reciprocal of \frac{2y^{7}}{x}.

$$\frac{27x}{2y^{7}}$$