Solve (y^{3}x^{2})(-3x^{2}y^{-3})^-3(2x^-2y^5)^2

$$( y ^ { 3 } x ^ { 2 } ) ( - 3 x ^ { 2 } y ^ { - 3 } ) ^ { - 3 } ( 2 x ^ { - 2 } y ^ { 5 } ) ^ { 2 }$$

Evaluate

$$-\frac{4y^{22}}{27x^{8}}$$

Solution Steps

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

$$y^{3}x^{2}\left(-3\right)^{-3}\left(x^{2}\right)^{-3}\left(y^{-3}\right)^{-3}\times \left(2x^{-2}y^{5}\right)^{2}$$

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

$$y^{3}x^{2}\left(-3\right)^{-3}x^{-6}\left(y^{-3}\right)^{-3}\times \left(2x^{-2}y^{5}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -3 to get 9.

$$y^{3}x^{2}\left(-3\right)^{-3}x^{-6}y^{9}\times \left(2x^{-2}y^{5}\right)^{2}$$

Calculate -3 to the power of -3 and get -\frac{1}{27}.

$$y^{3}x^{2}\left(-\frac{1}{27}\right)x^{-6}y^{9}\times \left(2x^{-2}y^{5}\right)^{2}$$

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

$$y^{3}x^{-4}\left(-\frac{1}{27}\right)y^{9}\times \left(2x^{-2}y^{5}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 3 and 9 to get 12.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times \left(2x^{-2}y^{5}\right)^{2}$$

Expand \left(2x^{-2}y^{5}\right)^{2}.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 2^{2}\left(x^{-2}\right)^{2}\left(y^{5}\right)^{2}$$

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

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 2^{2}x^{-4}\left(y^{5}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 2^{2}x^{-4}y^{10}$$

Calculate 2 to the power of 2 and get 4.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 4x^{-4}y^{10}$$

Multiply -\frac{1}{27} and 4 to get -\frac{4}{27}.

$$y^{12}x^{-4}\left(-\frac{4}{27}\right)x^{-4}y^{10}$$

To multiply powers of the same base, add their exponents. Add -4 and -4 to get -8.

$$y^{12}x^{-8}\left(-\frac{4}{27}\right)y^{10}$$

To multiply powers of the same base, add their exponents. Add 12 and 10 to get 22.

$$y^{22}x^{-8}\left(-\frac{4}{27}\right)$$

Expand

$$-\frac{4y^{22}}{27x^{8}}$$

Solution Steps

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

$$y^{3}x^{2}\left(-3\right)^{-3}\left(x^{2}\right)^{-3}\left(y^{-3}\right)^{-3}\times \left(2x^{-2}y^{5}\right)^{2}$$

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

$$y^{3}x^{2}\left(-3\right)^{-3}x^{-6}\left(y^{-3}\right)^{-3}\times \left(2x^{-2}y^{5}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -3 to get 9.

$$y^{3}x^{2}\left(-3\right)^{-3}x^{-6}y^{9}\times \left(2x^{-2}y^{5}\right)^{2}$$

Calculate -3 to the power of -3 and get -\frac{1}{27}.

$$y^{3}x^{2}\left(-\frac{1}{27}\right)x^{-6}y^{9}\times \left(2x^{-2}y^{5}\right)^{2}$$

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

$$y^{3}x^{-4}\left(-\frac{1}{27}\right)y^{9}\times \left(2x^{-2}y^{5}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 3 and 9 to get 12.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times \left(2x^{-2}y^{5}\right)^{2}$$

Expand \left(2x^{-2}y^{5}\right)^{2}.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 2^{2}\left(x^{-2}\right)^{2}\left(y^{5}\right)^{2}$$

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

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 2^{2}x^{-4}\left(y^{5}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 2^{2}x^{-4}y^{10}$$

Calculate 2 to the power of 2 and get 4.

$$y^{12}x^{-4}\left(-\frac{1}{27}\right)\times 4x^{-4}y^{10}$$

Multiply -\frac{1}{27} and 4 to get -\frac{4}{27}.

$$y^{12}x^{-4}\left(-\frac{4}{27}\right)x^{-4}y^{10}$$

To multiply powers of the same base, add their exponents. Add -4 and -4 to get -8.

$$y^{12}x^{-8}\left(-\frac{4}{27}\right)y^{10}$$

To multiply powers of the same base, add their exponents. Add 12 and 10 to get 22.

$$y^{22}x^{-8}\left(-\frac{4}{27}\right)$$