Solve frac{(6x^{3}y^{-4})^-2}{(3x^2y^5)^-3}

$$\frac { ( 6 x ^ { 3 } y ^ { - 4 } ) ^ { - 2 } } { ( 3 x ^ { 2 } y ^ { 5 } ) ^ { - 3 } }$$

Evaluate

$$\frac{3y^{23}}{4}$$

Solution Steps

Expand \left(6x^{3}y^{-4}\right)^{-2}.

$$\frac{6^{-2}\left(x^{3}\right)^{-2}\left(y^{-4}\right)^{-2}}{\left(3x^{2}y^{5}\right)^{-3}}$$

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

$$\frac{6^{-2}x^{-6}\left(y^{-4}\right)^{-2}}{\left(3x^{2}y^{5}\right)^{-3}}$$

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

$$\frac{6^{-2}x^{-6}y^{8}}{\left(3x^{2}y^{5}\right)^{-3}}$$

Calculate 6 to the power of -2 and get \frac{1}{36}.

$$\frac{\frac{1}{36}x^{-6}y^{8}}{\left(3x^{2}y^{5}\right)^{-3}}$$

Expand \left(3x^{2}y^{5}\right)^{-3}.

$$\frac{\frac{1}{36}x^{-6}y^{8}}{3^{-3}\left(x^{2}\right)^{-3}\left(y^{5}\right)^{-3}}$$

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

$$\frac{\frac{1}{36}x^{-6}y^{8}}{3^{-3}x^{-6}\left(y^{5}\right)^{-3}}$$

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

$$\frac{\frac{1}{36}x^{-6}y^{8}}{3^{-3}x^{-6}y^{-15}}$$

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

$$\frac{\frac{1}{36}x^{-6}y^{8}}{\frac{1}{27}x^{-6}y^{-15}}$$

Cancel out x^{-6} in both numerator and denominator.

$$\frac{\frac{1}{36}y^{8}}{\frac{1}{27}y^{-15}}$$

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

$$\frac{\frac{1}{36}y^{23}}{\frac{1}{27}}$$

Divide \frac{1}{36}y^{23} by \frac{1}{27} by multiplying \frac{1}{36}y^{23} by the reciprocal of \frac{1}{27}.

$$\frac{1}{36}y^{23}\times 27$$

Multiply \frac{1}{36} and 27 to get \frac{3}{4}.

$$\frac{3}{4}y^{23}$$

Expand

$$\frac{3y^{23}}{4}$$

Solution Steps

Expand \left(6x^{3}y^{-4}\right)^{-2}.

$$\frac{6^{-2}\left(x^{3}\right)^{-2}\left(y^{-4}\right)^{-2}}{\left(3x^{2}y^{5}\right)^{-3}}$$

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

$$\frac{6^{-2}x^{-6}\left(y^{-4}\right)^{-2}}{\left(3x^{2}y^{5}\right)^{-3}}$$

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

$$\frac{6^{-2}x^{-6}y^{8}}{\left(3x^{2}y^{5}\right)^{-3}}$$

Calculate 6 to the power of -2 and get \frac{1}{36}.

$$\frac{\frac{1}{36}x^{-6}y^{8}}{\left(3x^{2}y^{5}\right)^{-3}}$$

Expand \left(3x^{2}y^{5}\right)^{-3}.

$$\frac{\frac{1}{36}x^{-6}y^{8}}{3^{-3}\left(x^{2}\right)^{-3}\left(y^{5}\right)^{-3}}$$

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

$$\frac{\frac{1}{36}x^{-6}y^{8}}{3^{-3}x^{-6}\left(y^{5}\right)^{-3}}$$

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

$$\frac{\frac{1}{36}x^{-6}y^{8}}{3^{-3}x^{-6}y^{-15}}$$

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

$$\frac{\frac{1}{36}x^{-6}y^{8}}{\frac{1}{27}x^{-6}y^{-15}}$$

Cancel out x^{-6} in both numerator and denominator.

$$\frac{\frac{1}{36}y^{8}}{\frac{1}{27}y^{-15}}$$

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

$$\frac{\frac{1}{36}y^{23}}{\frac{1}{27}}$$

Divide \frac{1}{36}y^{23} by \frac{1}{27} by multiplying \frac{1}{36}y^{23} by the reciprocal of \frac{1}{27}.

$$\frac{1}{36}y^{23}\times 27$$

Multiply \frac{1}{36} and 27 to get \frac{3}{4}.

$$\frac{3}{4}y^{23}$$