Solve 25/left(x^-2yright)^4*{left(5x^2y^-3/3{x^-5{y}^-2}right)}^-2

$$\frac{ 25 }{ { \left( { x }^{ -2 } y \right) }^{ 4 } } \times { \left( \frac{ 5 { x }^{ 2 } { y }^{ -3 } }{ 3 { x }^{ -5 } { y }^{ -2 } } \right) }^{ -2 }$$

Evaluate

$$\frac{9}{y^{2}x^{6}}$$

Solution Steps

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

$$\frac{25}{\left(x^{-2}\right)^{4}y^{4}}\times \left(\frac{5x^{2}y^{-3}}{3x^{-5}y^{-2}}\right)^{-2}$$

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

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5x^{2}y^{-3}}{3x^{-5}y^{-2}}\right)^{-2}$$

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

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5y^{-3}x^{7}}{3y^{-2}}\right)^{-2}$$

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

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5x^{7}}{3y^{1}}\right)^{-2}$$

Calculate y to the power of 1 and get y.

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5x^{7}}{3y}\right)^{-2}$$

To raise \frac{5x^{7}}{3y} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{25}{x^{-8}y^{4}}\times \frac{\left(5x^{7}\right)^{-2}}{\left(3y\right)^{-2}}$$

Multiply \frac{25}{x^{-8}y^{4}} times \frac{\left(5x^{7}\right)^{-2}}{\left(3y\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25\times \left(5x^{7}\right)^{-2}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

Expand \left(5x^{7}\right)^{-2}.

$$\frac{25\times 5^{-2}\left(x^{7}\right)^{-2}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

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

$$\frac{25\times 5^{-2}x^{-14}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{25\times \frac{1}{25}x^{-14}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

Multiply 25 and \frac{1}{25} to get 1.

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

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

$$\frac{x^{-14}}{x^{-8}y^{4}\times 3^{-2}y^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{x^{-14}}{x^{-8}y^{4}\times \frac{1}{9}y^{-2}}$$

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

$$\frac{x^{-14}}{x^{-8}y^{2}\times \frac{1}{9}}$$

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

$$\frac{1}{\frac{1}{9}y^{2}x^{6}}$$

Expand

$$\frac{9}{y^{2}x^{6}}$$

Solution Steps

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

$$\frac{25}{\left(x^{-2}\right)^{4}y^{4}}\times \left(\frac{5x^{2}y^{-3}}{3x^{-5}y^{-2}}\right)^{-2}$$

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

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5x^{2}y^{-3}}{3x^{-5}y^{-2}}\right)^{-2}$$

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

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5y^{-3}x^{7}}{3y^{-2}}\right)^{-2}$$

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

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5x^{7}}{3y^{1}}\right)^{-2}$$

Calculate y to the power of 1 and get y.

$$\frac{25}{x^{-8}y^{4}}\times \left(\frac{5x^{7}}{3y}\right)^{-2}$$

To raise \frac{5x^{7}}{3y} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{25}{x^{-8}y^{4}}\times \frac{\left(5x^{7}\right)^{-2}}{\left(3y\right)^{-2}}$$

Multiply \frac{25}{x^{-8}y^{4}} times \frac{\left(5x^{7}\right)^{-2}}{\left(3y\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25\times \left(5x^{7}\right)^{-2}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

Expand \left(5x^{7}\right)^{-2}.

$$\frac{25\times 5^{-2}\left(x^{7}\right)^{-2}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

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

$$\frac{25\times 5^{-2}x^{-14}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{25\times \frac{1}{25}x^{-14}}{x^{-8}y^{4}\times \left(3y\right)^{-2}}$$

Multiply 25 and \frac{1}{25} to get 1.

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

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

$$\frac{x^{-14}}{x^{-8}y^{4}\times 3^{-2}y^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{x^{-14}}{x^{-8}y^{4}\times \frac{1}{9}y^{-2}}$$

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

$$\frac{x^{-14}}{x^{-8}y^{2}\times \frac{1}{9}}$$

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

$$\frac{1}{\frac{1}{9}y^{2}x^{6}}$$