Solve {[frac{(2ay^{-1})^{-1}(frac{y^{3}}{2ya^{-2}})^{2}}{(2a^{-1}y)^-2}]^-2ay^-5}^-1

$$\{ [ \frac { ( 2 a y ^ { - 1 } ) ^ { - 1 } ( \frac { y ^ { 3 } } { 2 y a ^ { - 2 } } ) ^ { 2 } } { ( 2 a ^ { - 1 } y ) ^ { - 2 } } ] ^ { - 2 } a y ^ { - 5 } \} ^ { - 1 }$$

Evaluate

$$\frac{ay^{19}}{4}$$

Solution Steps

Expand \left(2ay^{-1}\right)^{-1}.

$$\left(\left(\frac{2^{-1}a^{-1}\left(y^{-1}\right)^{-1}\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -1 to get 1.

$$\left(\left(\frac{2^{-1}a^{-1}y^{1}\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{1}{2}a^{-1}y^{1}\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Calculate y to the power of 1 and get y.

$$\left(\left(\frac{\frac{1}{2}a^{-1}y\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Cancel out y in both numerator and denominator.

$$\left(\left(\frac{\frac{1}{2}a^{-1}y\times \left(\frac{y^{2}}{2a^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{1}{2}a^{-1}y\times \frac{\left(y^{2}\right)^{2}}{\left(2a^{-2}\right)^{2}}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Multiply \frac{1}{2} times \frac{\left(y^{2}\right)^{2}}{\left(2a^{-2}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Expand \left(2a^{-1}y\right)^{-2}.

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{2^{-2}\left(a^{-1}\right)^{-2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -2 to get 2.

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{2^{-2}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{y^{4}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{y^{4}}{2\times 2^{2}\left(a^{-2}\right)^{2}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{y^{4}}{2\times 2^{2}a^{-4}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Calculate 2 to the power of 2 and get 4.

$$\left(\left(\frac{\frac{y^{4}}{2\times 4a^{-4}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Multiply 2 and 4 to get 8.

$$\left(\left(\frac{\frac{y^{4}}{8a^{-4}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Express \frac{y^{4}}{8a^{-4}}a^{-1} as a single fraction.

$$\left(\left(\frac{\frac{y^{4}a^{-1}}{8a^{-4}}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{a^{3}y^{4}}{8}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Express \frac{a^{3}y^{4}}{8}y as a single fraction.

$$\left(\left(\frac{\frac{a^{3}y^{4}y}{8}}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Express \frac{\frac{a^{3}y^{4}y}{8}}{\frac{1}{4}a^{2}y^{-2}} as a single fraction.

$$\left(\left(\frac{a^{3}y^{4}y}{8\times \frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Cancel out a^{2} in both numerator and denominator.

$$\left(\left(\frac{ayy^{4}}{\frac{1}{4}\times 8y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{ayy^{6}}{\frac{1}{4}\times 8}\right)^{-2}ay^{-5}\right)^{-1}$$

To raise \frac{ayy^{6}}{\frac{1}{4}\times 8} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\left(ayy^{6}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}ay^{-5}\right)^{-1}$$

Expand \left(\frac{\left(ayy^{6}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}ay^{-5}\right)^{-1}.

$$\left(\frac{\left(ayy^{6}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}\left(y^{-5}\right)^{-1}$$

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

$$\left(\frac{\left(ay^{7}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}\left(y^{-5}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -1 to get 5.

$$\left(\frac{\left(ay^{7}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}y^{5}$$

Expand \left(ay^{7}\right)^{-2}.

$$\left(\frac{a^{-2}\left(y^{7}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}y^{5}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$\left(\frac{a^{-2}y^{-14}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}y^{5}$$

Multiply \frac{1}{4} and 8 to get 2.

$$\left(\frac{a^{-2}y^{-14}}{2^{-2}}\right)^{-1}a^{-1}y^{5}$$

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

$$\left(\frac{a^{-2}y^{-14}}{\frac{1}{4}}\right)^{-1}a^{-1}y^{5}$$

Divide a^{-2}y^{-14} by \frac{1}{4} by multiplying a^{-2}y^{-14} by the reciprocal of \frac{1}{4}.

$$\left(a^{-2}y^{-14}\times 4\right)^{-1}a^{-1}y^{5}$$

Expand \left(a^{-2}y^{-14}\times 4\right)^{-1}.

$$\left(a^{-2}\right)^{-1}\left(y^{-14}\right)^{-1}\times 4^{-1}a^{-1}y^{5}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$a^{2}\left(y^{-14}\right)^{-1}\times 4^{-1}a^{-1}y^{5}$$

To raise a power to another power, multiply the exponents. Multiply -14 and -1 to get 14.

$$a^{2}y^{14}\times 4^{-1}a^{-1}y^{5}$$

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

$$a^{2}y^{14}\times \frac{1}{4}a^{-1}y^{5}$$

To multiply powers of the same base, add their exponents. Add 2 and -1 to get 1.

$$a^{1}y^{14}\times \frac{1}{4}y^{5}$$

To multiply powers of the same base, add their exponents. Add 14 and 5 to get 19.

$$a^{1}y^{19}\times \frac{1}{4}$$

Calculate a to the power of 1 and get a.

$$ay^{19}\times \frac{1}{4}$$

Expand

$$\frac{ay^{19}}{4}$$

Solution Steps

Expand \left(2ay^{-1}\right)^{-1}.

$$\left(\left(\frac{2^{-1}a^{-1}\left(y^{-1}\right)^{-1}\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -1 to get 1.

$$\left(\left(\frac{2^{-1}a^{-1}y^{1}\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{1}{2}a^{-1}y^{1}\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Calculate y to the power of 1 and get y.

$$\left(\left(\frac{\frac{1}{2}a^{-1}y\times \left(\frac{y^{3}}{2ya^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Cancel out y in both numerator and denominator.

$$\left(\left(\frac{\frac{1}{2}a^{-1}y\times \left(\frac{y^{2}}{2a^{-2}}\right)^{2}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{1}{2}a^{-1}y\times \frac{\left(y^{2}\right)^{2}}{\left(2a^{-2}\right)^{2}}}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Multiply \frac{1}{2} times \frac{\left(y^{2}\right)^{2}}{\left(2a^{-2}\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{\left(2a^{-1}y\right)^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Expand \left(2a^{-1}y\right)^{-2}.

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{2^{-2}\left(a^{-1}\right)^{-2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -2 to get 2.

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{2^{-2}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{\left(y^{2}\right)^{2}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{y^{4}}{2\times \left(2a^{-2}\right)^{2}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{y^{4}}{2\times 2^{2}\left(a^{-2}\right)^{2}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{y^{4}}{2\times 2^{2}a^{-4}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Calculate 2 to the power of 2 and get 4.

$$\left(\left(\frac{\frac{y^{4}}{2\times 4a^{-4}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Multiply 2 and 4 to get 8.

$$\left(\left(\frac{\frac{y^{4}}{8a^{-4}}a^{-1}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Express \frac{y^{4}}{8a^{-4}}a^{-1} as a single fraction.

$$\left(\left(\frac{\frac{y^{4}a^{-1}}{8a^{-4}}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{a^{3}y^{4}}{8}y}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Express \frac{a^{3}y^{4}}{8}y as a single fraction.

$$\left(\left(\frac{\frac{a^{3}y^{4}y}{8}}{\frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Express \frac{\frac{a^{3}y^{4}y}{8}}{\frac{1}{4}a^{2}y^{-2}} as a single fraction.

$$\left(\left(\frac{a^{3}y^{4}y}{8\times \frac{1}{4}a^{2}y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

Cancel out a^{2} in both numerator and denominator.

$$\left(\left(\frac{ayy^{4}}{\frac{1}{4}\times 8y^{-2}}\right)^{-2}ay^{-5}\right)^{-1}$$

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

$$\left(\left(\frac{ayy^{6}}{\frac{1}{4}\times 8}\right)^{-2}ay^{-5}\right)^{-1}$$

To raise \frac{ayy^{6}}{\frac{1}{4}\times 8} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\left(ayy^{6}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}ay^{-5}\right)^{-1}$$

Expand \left(\frac{\left(ayy^{6}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}ay^{-5}\right)^{-1}.

$$\left(\frac{\left(ayy^{6}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}\left(y^{-5}\right)^{-1}$$

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

$$\left(\frac{\left(ay^{7}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}\left(y^{-5}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -1 to get 5.

$$\left(\frac{\left(ay^{7}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}y^{5}$$

Expand \left(ay^{7}\right)^{-2}.

$$\left(\frac{a^{-2}\left(y^{7}\right)^{-2}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}y^{5}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$\left(\frac{a^{-2}y^{-14}}{\left(\frac{1}{4}\times 8\right)^{-2}}\right)^{-1}a^{-1}y^{5}$$

Multiply \frac{1}{4} and 8 to get 2.

$$\left(\frac{a^{-2}y^{-14}}{2^{-2}}\right)^{-1}a^{-1}y^{5}$$

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

$$\left(\frac{a^{-2}y^{-14}}{\frac{1}{4}}\right)^{-1}a^{-1}y^{5}$$

Divide a^{-2}y^{-14} by \frac{1}{4} by multiplying a^{-2}y^{-14} by the reciprocal of \frac{1}{4}.

$$\left(a^{-2}y^{-14}\times 4\right)^{-1}a^{-1}y^{5}$$

Expand \left(a^{-2}y^{-14}\times 4\right)^{-1}.

$$\left(a^{-2}\right)^{-1}\left(y^{-14}\right)^{-1}\times 4^{-1}a^{-1}y^{5}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$a^{2}\left(y^{-14}\right)^{-1}\times 4^{-1}a^{-1}y^{5}$$

To raise a power to another power, multiply the exponents. Multiply -14 and -1 to get 14.

$$a^{2}y^{14}\times 4^{-1}a^{-1}y^{5}$$

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

$$a^{2}y^{14}\times \frac{1}{4}a^{-1}y^{5}$$

To multiply powers of the same base, add their exponents. Add 2 and -1 to get 1.

$$a^{1}y^{14}\times \frac{1}{4}y^{5}$$

To multiply powers of the same base, add their exponents. Add 14 and 5 to get 19.

$$a^{1}y^{19}\times \frac{1}{4}$$

Calculate a to the power of 1 and get a.

$$ay^{19}\times \frac{1}{4}$$