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

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

Evaluate

$$\frac{20y^{11}}{9x^{6}}$$

Solution Steps

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

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

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

$$\frac{15y\times 3^{-3}x^{-12}\left(y^{-2}\right)^{-3}}{\left(2x^{3}y^{2}\right)^{-2}}$$

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

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

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

$$\frac{15y\times \frac{1}{27}x^{-12}y^{6}}{\left(2x^{3}y^{2}\right)^{-2}}$$

Multiply 15 and \frac{1}{27} to get \frac{5}{9}.

$$\frac{\frac{5}{9}yx^{-12}y^{6}}{\left(2x^{3}y^{2}\right)^{-2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{\frac{5}{9}y^{7}x^{-12}}{\left(2x^{3}y^{2}\right)^{-2}}$$

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

$$\frac{\frac{5}{9}y^{7}x^{-12}}{2^{-2}\left(x^{3}\right)^{-2}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{5}{9}y^{7}x^{-12}}{2^{-2}x^{-6}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{5}{9}y^{7}x^{-12}}{2^{-2}x^{-6}y^{-4}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{5}{9}y^{7}x^{-12}}{\frac{1}{4}x^{-6}y^{-4}}$$

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

$$\frac{\frac{5}{9}x^{-12}y^{11}}{\frac{1}{4}x^{-6}}$$

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

$$\frac{\frac{5}{9}y^{11}}{\frac{1}{4}x^{6}}$$

Expand

$$\frac{20y^{11}}{9x^{6}}$$

Solution Steps

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

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

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

$$\frac{15y\times 3^{-3}x^{-12}\left(y^{-2}\right)^{-3}}{\left(2x^{3}y^{2}\right)^{-2}}$$

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

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

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

$$\frac{15y\times \frac{1}{27}x^{-12}y^{6}}{\left(2x^{3}y^{2}\right)^{-2}}$$

Multiply 15 and \frac{1}{27} to get \frac{5}{9}.

$$\frac{\frac{5}{9}yx^{-12}y^{6}}{\left(2x^{3}y^{2}\right)^{-2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$\frac{\frac{5}{9}y^{7}x^{-12}}{\left(2x^{3}y^{2}\right)^{-2}}$$

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

$$\frac{\frac{5}{9}y^{7}x^{-12}}{2^{-2}\left(x^{3}\right)^{-2}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{5}{9}y^{7}x^{-12}}{2^{-2}x^{-6}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{5}{9}y^{7}x^{-12}}{2^{-2}x^{-6}y^{-4}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{5}{9}y^{7}x^{-12}}{\frac{1}{4}x^{-6}y^{-4}}$$

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

$$\frac{\frac{5}{9}x^{-12}y^{11}}{\frac{1}{4}x^{-6}}$$

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

$$\frac{\frac{5}{9}y^{11}}{\frac{1}{4}x^{6}}$$