Solve (frac{(frac{s^{2}t^{5}}{s^{4}t^{3}})^{-1}}{(frac{2s^{3}t^{4}}{s^{2}})^{2}})^2((12/5^2)^-1/(25+4)^2)

$$( \frac { ( \frac { s ^ { 2 } t ^ { 5 } } { s ^ { 4 } t ^ { 3 } } ) ^ { - 1 } } { ( \frac { 2 s ^ { 3 } t ^ { 4 } } { s ^ { 2 } } ) ^ { 2 } } ) ^ { 2 } ( \frac { ( \frac { 12 } { 5 ^ { 2 } } ) ^ { - 1 } } { ( 25 + 4 ) ^ { 2 } } )$$

Evaluate

$$\frac{25}{161472t^{20}}$$

Solution Steps

Cancel out s^{2}t^{3} in both numerator and denominator.

$$\left(\frac{\left(\frac{t^{2}}{s^{2}}\right)^{-1}}{\left(\frac{2s^{3}t^{4}}{s^{2}}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{\left(\frac{2s^{3}t^{4}}{s^{2}}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{\left(2st^{4}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Expand \left(2st^{4}\right)^{2}.

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{2^{2}s^{2}\left(t^{4}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{2^{2}s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{4s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Express \frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{4s^{2}t^{8}} as a single fraction.

$$\left(\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}\times 4s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\left(t^{2}\right)^{-1}}{s^{-2}\times 4s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Multiply s^{-2} and s^{2} to get 1.

$$\left(\frac{\left(t^{2}\right)^{-1}}{4t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

To raise \frac{\left(t^{2}\right)^{-1}}{4t^{8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\left(\frac{12}{25}\right)^{-1}}{\left(25+4\right)^{2}}$$

Calculate \frac{12}{25} to the power of -1 and get \frac{25}{12}.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\frac{25}{12}}{\left(25+4\right)^{2}}$$

Add 25 and 4 to get 29.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\frac{25}{12}}{29^{2}}$$

Calculate 29 to the power of 2 and get 841.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\frac{25}{12}}{841}$$

Express \frac{\frac{25}{12}}{841} as a single fraction.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{25}{12\times 841}$$

Multiply 12 and 841 to get 10092.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{25}{10092}$$

Multiply \frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}} times \frac{25}{10092} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}\times 25}{\left(4t^{8}\right)^{2}\times 10092}$$

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

$$\frac{\left(t^{-2}\right)^{2}\times 25}{\left(4t^{8}\right)^{2}\times 10092}$$

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

$$\frac{t^{-4}\times 25}{\left(4t^{8}\right)^{2}\times 10092}$$

Expand \left(4t^{8}\right)^{2}.

$$\frac{t^{-4}\times 25}{4^{2}\left(t^{8}\right)^{2}\times 10092}$$

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

$$\frac{t^{-4}\times 25}{4^{2}t^{16}\times 10092}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{t^{-4}\times 25}{16t^{16}\times 10092}$$

Multiply 16 and 10092 to get 161472.

$$\frac{t^{-4}\times 25}{161472t^{16}}$$

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

$$\frac{25}{161472t^{20}}$$

Expand

$$\frac{25}{161472t^{20}}$$

Solution Steps

Cancel out s^{2}t^{3} in both numerator and denominator.

$$\left(\frac{\left(\frac{t^{2}}{s^{2}}\right)^{-1}}{\left(\frac{2s^{3}t^{4}}{s^{2}}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{\left(\frac{2s^{3}t^{4}}{s^{2}}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{\left(2st^{4}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Expand \left(2st^{4}\right)^{2}.

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{2^{2}s^{2}\left(t^{4}\right)^{2}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{2^{2}s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\left(\frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{4s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Express \frac{\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}}}{4s^{2}t^{8}} as a single fraction.

$$\left(\frac{\left(t^{2}\right)^{-1}}{\left(s^{2}\right)^{-1}\times 4s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

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

$$\left(\frac{\left(t^{2}\right)^{-1}}{s^{-2}\times 4s^{2}t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Multiply s^{-2} and s^{2} to get 1.

$$\left(\frac{\left(t^{2}\right)^{-1}}{4t^{8}}\right)^{2}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

To raise \frac{\left(t^{2}\right)^{-1}}{4t^{8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\left(\frac{12}{5^{2}}\right)^{-1}}{\left(25+4\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\left(\frac{12}{25}\right)^{-1}}{\left(25+4\right)^{2}}$$

Calculate \frac{12}{25} to the power of -1 and get \frac{25}{12}.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\frac{25}{12}}{\left(25+4\right)^{2}}$$

Add 25 and 4 to get 29.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\frac{25}{12}}{29^{2}}$$

Calculate 29 to the power of 2 and get 841.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{\frac{25}{12}}{841}$$

Express \frac{\frac{25}{12}}{841} as a single fraction.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{25}{12\times 841}$$

Multiply 12 and 841 to get 10092.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}}\times \frac{25}{10092}$$

Multiply \frac{\left(\left(t^{2}\right)^{-1}\right)^{2}}{\left(4t^{8}\right)^{2}} times \frac{25}{10092} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\left(t^{2}\right)^{-1}\right)^{2}\times 25}{\left(4t^{8}\right)^{2}\times 10092}$$

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

$$\frac{\left(t^{-2}\right)^{2}\times 25}{\left(4t^{8}\right)^{2}\times 10092}$$

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

$$\frac{t^{-4}\times 25}{\left(4t^{8}\right)^{2}\times 10092}$$

Expand \left(4t^{8}\right)^{2}.

$$\frac{t^{-4}\times 25}{4^{2}\left(t^{8}\right)^{2}\times 10092}$$

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

$$\frac{t^{-4}\times 25}{4^{2}t^{16}\times 10092}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{t^{-4}\times 25}{16t^{16}\times 10092}$$

Multiply 16 and 10092 to get 161472.

$$\frac{t^{-4}\times 25}{161472t^{16}}$$

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

$$\frac{25}{161472t^{20}}$$