Solve frac{(15x^{2}y^{-3})^-4}{(25x^3y^-6)^-2}=

$$\frac { ( 15 x ^ { 2 } y ^ { - 3 } ) ^ { - 4 } } { ( 25 x ^ { 3 } y ^ { - 6 } ) ^ { - 2 } } =$$

Evaluate

$$\frac{1}{81x^{2}}$$

Solution Steps

Expand \left(15x^{2}y^{-3}\right)^{-4}.

$$\frac{15^{-4}\left(x^{2}\right)^{-4}\left(y^{-3}\right)^{-4}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

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

$$\frac{15^{-4}x^{-8}\left(y^{-3}\right)^{-4}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -4 to get 12.

$$\frac{15^{-4}x^{-8}y^{12}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

Calculate 15 to the power of -4 and get \frac{1}{50625}.

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

Expand \left(25x^{3}y^{-6}\right)^{-2}.

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{25^{-2}\left(x^{3}\right)^{-2}\left(y^{-6}\right)^{-2}}$$

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

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{25^{-2}x^{-6}\left(y^{-6}\right)^{-2}}$$

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

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{25^{-2}x^{-6}y^{12}}$$

Calculate 25 to the power of -2 and get \frac{1}{625}.

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{\frac{1}{625}x^{-6}y^{12}}$$

Cancel out y^{12} in both numerator and denominator.

$$\frac{\frac{1}{50625}x^{-8}}{\frac{1}{625}x^{-6}}$$

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

$$\frac{\frac{1}{50625}}{\frac{1}{625}x^{2}}$$

Express \frac{\frac{1}{50625}}{\frac{1}{625}x^{2}} as a single fraction.

$$\frac{1}{50625\times \frac{1}{625}x^{2}}$$

Multiply 50625 and \frac{1}{625} to get 81.

$$\frac{1}{81x^{2}}$$

Expand

$$\frac{1}{81x^{2}}$$

Solution Steps

Expand \left(15x^{2}y^{-3}\right)^{-4}.

$$\frac{15^{-4}\left(x^{2}\right)^{-4}\left(y^{-3}\right)^{-4}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

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

$$\frac{15^{-4}x^{-8}\left(y^{-3}\right)^{-4}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -4 to get 12.

$$\frac{15^{-4}x^{-8}y^{12}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

Calculate 15 to the power of -4 and get \frac{1}{50625}.

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{\left(25x^{3}y^{-6}\right)^{-2}}$$

Expand \left(25x^{3}y^{-6}\right)^{-2}.

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{25^{-2}\left(x^{3}\right)^{-2}\left(y^{-6}\right)^{-2}}$$

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

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{25^{-2}x^{-6}\left(y^{-6}\right)^{-2}}$$

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

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{25^{-2}x^{-6}y^{12}}$$

Calculate 25 to the power of -2 and get \frac{1}{625}.

$$\frac{\frac{1}{50625}x^{-8}y^{12}}{\frac{1}{625}x^{-6}y^{12}}$$

Cancel out y^{12} in both numerator and denominator.

$$\frac{\frac{1}{50625}x^{-8}}{\frac{1}{625}x^{-6}}$$

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

$$\frac{\frac{1}{50625}}{\frac{1}{625}x^{2}}$$

Express \frac{\frac{1}{50625}}{\frac{1}{625}x^{2}} as a single fraction.

$$\frac{1}{50625\times \frac{1}{625}x^{2}}$$

Multiply 50625 and \frac{1}{625} to get 81.

$$\frac{1}{81x^{2}}$$