Solve (frac{5xy^{2}}{3x^{-2}y^{-3}})^{3}(frac{3x^{-3}y^-4}{5x^3y^7})^2

$$( \frac { 5 x y ^ { 2 } } { 3 x ^ { - 2 } y ^ { - 3 } } ) ^ { 3 } ( \frac { 3 x ^ { - 3 } y ^ { - 4 } } { 5 x ^ { 3 } y ^ { 7 } } ) ^ { 2 }$$

Evaluate

$$\frac{5}{3x^{3}y^{7}}$$

Solution Steps

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

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

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

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

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

$$\frac{\left(5x^{3}y^{5}\right)^{3}}{3^{3}}\times \left(\frac{3}{5x^{6}y^{11}}\right)^{2}$$

To raise \frac{3}{5x^{6}y^{11}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(5x^{3}y^{5}\right)^{3}}{3^{3}}\times \frac{3^{2}}{\left(5x^{6}y^{11}\right)^{2}}$$

Multiply \frac{\left(5x^{3}y^{5}\right)^{3}}{3^{3}} times \frac{3^{2}}{\left(5x^{6}y^{11}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(5x^{3}y^{5}\right)^{3}\times 3^{2}}{3^{3}\times \left(5x^{6}y^{11}\right)^{2}}$$

Cancel out 3^{2} in both numerator and denominator.

$$\frac{\left(5x^{3}y^{5}\right)^{3}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

Expand \left(5x^{3}y^{5}\right)^{3}.

$$\frac{5^{3}\left(x^{3}\right)^{3}\left(y^{5}\right)^{3}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

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

$$\frac{5^{3}x^{9}\left(y^{5}\right)^{3}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{5^{3}x^{9}y^{15}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

Calculate 5 to the power of 3 and get 125.

$$\frac{125x^{9}y^{15}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

Expand \left(5x^{6}y^{11}\right)^{2}.

$$\frac{125x^{9}y^{15}}{3\times 5^{2}\left(x^{6}\right)^{2}\left(y^{11}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{125x^{9}y^{15}}{3\times 5^{2}x^{12}\left(y^{11}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 11 and 2 to get 22.

$$\frac{125x^{9}y^{15}}{3\times 5^{2}x^{12}y^{22}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{125x^{9}y^{15}}{3\times 25x^{12}y^{22}}$$

Multiply 3 and 25 to get 75.

$$\frac{125x^{9}y^{15}}{75x^{12}y^{22}}$$

Cancel out 25x^{9}y^{15} in both numerator and denominator.

$$\frac{5}{3x^{3}y^{7}}$$

Expand

$$\frac{5}{3x^{3}y^{7}}$$

Solution Steps

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

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

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

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

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

$$\frac{\left(5x^{3}y^{5}\right)^{3}}{3^{3}}\times \left(\frac{3}{5x^{6}y^{11}}\right)^{2}$$

To raise \frac{3}{5x^{6}y^{11}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(5x^{3}y^{5}\right)^{3}}{3^{3}}\times \frac{3^{2}}{\left(5x^{6}y^{11}\right)^{2}}$$

Multiply \frac{\left(5x^{3}y^{5}\right)^{3}}{3^{3}} times \frac{3^{2}}{\left(5x^{6}y^{11}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(5x^{3}y^{5}\right)^{3}\times 3^{2}}{3^{3}\times \left(5x^{6}y^{11}\right)^{2}}$$

Cancel out 3^{2} in both numerator and denominator.

$$\frac{\left(5x^{3}y^{5}\right)^{3}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

Expand \left(5x^{3}y^{5}\right)^{3}.

$$\frac{5^{3}\left(x^{3}\right)^{3}\left(y^{5}\right)^{3}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

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

$$\frac{5^{3}x^{9}\left(y^{5}\right)^{3}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{5^{3}x^{9}y^{15}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

Calculate 5 to the power of 3 and get 125.

$$\frac{125x^{9}y^{15}}{3\times \left(5x^{6}y^{11}\right)^{2}}$$

Expand \left(5x^{6}y^{11}\right)^{2}.

$$\frac{125x^{9}y^{15}}{3\times 5^{2}\left(x^{6}\right)^{2}\left(y^{11}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{125x^{9}y^{15}}{3\times 5^{2}x^{12}\left(y^{11}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 11 and 2 to get 22.

$$\frac{125x^{9}y^{15}}{3\times 5^{2}x^{12}y^{22}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{125x^{9}y^{15}}{3\times 25x^{12}y^{22}}$$

Multiply 3 and 25 to get 75.

$$\frac{125x^{9}y^{15}}{75x^{12}y^{22}}$$

Cancel out 25x^{9}y^{15} in both numerator and denominator.

$$\frac{5}{3x^{3}y^{7}}$$