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

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

Evaluate

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

Solution Steps

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

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

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

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

Calculate 2 to the power of -4 and get \frac{1}{16}.

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{\left(4x^{3}y^{2}\right)^{-2}}$$

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

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{4^{-2}\left(x^{3}\right)^{-2}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{4^{-2}x^{-6}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{4^{-2}x^{-6}y^{-4}}$$

Calculate 4 to the power of -2 and get \frac{1}{16}.

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{\frac{1}{16}x^{-6}y^{-4}}$$

Cancel out y^{-4} in both numerator and denominator.

$$\frac{\frac{1}{16}x^{-8}}{\frac{1}{16}x^{-6}}$$

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

$$\frac{1}{\left(\frac{1}{16}\right)^{0}x^{2}}$$

Calculate \frac{1}{16} to the power of 0 and get 1.

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

For any term t, t\times 1=t and 1t=t.

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

Expand

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

Solution Steps

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

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

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

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

Calculate 2 to the power of -4 and get \frac{1}{16}.

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{\left(4x^{3}y^{2}\right)^{-2}}$$

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

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{4^{-2}\left(x^{3}\right)^{-2}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{4^{-2}x^{-6}\left(y^{2}\right)^{-2}}$$

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

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{4^{-2}x^{-6}y^{-4}}$$

Calculate 4 to the power of -2 and get \frac{1}{16}.

$$\frac{\frac{1}{16}x^{-8}y^{-4}}{\frac{1}{16}x^{-6}y^{-4}}$$

Cancel out y^{-4} in both numerator and denominator.

$$\frac{\frac{1}{16}x^{-8}}{\frac{1}{16}x^{-6}}$$

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

$$\frac{1}{\left(\frac{1}{16}\right)^{0}x^{2}}$$

Calculate \frac{1}{16} to the power of 0 and get 1.

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

For any term t, t\times 1=t and 1t=t.

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