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

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

Evaluate

$$\frac{x^{4}y^{16}}{4}$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Expand \left(xy^{2}\right)^{2}.

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and -4 to get -2.

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

Cancel out 2 in both numerator and denominator.

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

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

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

To raise \frac{2}{x^{2}y^{8}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Expand \left(x^{2}y^{8}\right)^{-2}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 8 and -2 to get -16.

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

Express \frac{\frac{1}{4}}{x^{-4}y^{-16}} as a single fraction.

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

Expand

$$\frac{x^{4}y^{16}}{4}$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Expand \left(xy^{2}\right)^{2}.

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and -4 to get -2.

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

Cancel out 2 in both numerator and denominator.

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

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

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

To raise \frac{2}{x^{2}y^{8}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Expand \left(x^{2}y^{8}\right)^{-2}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 8 and -2 to get -16.

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

Express \frac{\frac{1}{4}}{x^{-4}y^{-16}} as a single fraction.

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