Solve (frac{3^{-4}*5^{-1}}{3^{2}*5^{-3}})^{-1/2}(frac{3^{4}*5^3}{3^2*5^4})^-1

$$( \frac { 3 ^ { - 4 } \cdot 5 ^ { - 1 } } { 3 ^ { 2 } \cdot 5 ^ { - 3 } } ) ^ { - \frac { 1 } { 2 } } ( \frac { 3 ^ { 4 } \cdot 5 ^ { 3 } } { 3 ^ { 2 } \cdot 5 ^ { 4 } } ) ^ { - 1 }$$

Evaluate

$$3$$

Solution Steps

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

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

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

$$\left(\frac{5^{2}}{3^{6}}\right)^{-\frac{1}{2}}\times \left(\frac{3^{4}\times 5^{3}}{3^{2}\times 5^{4}}\right)^{-1}$$

Calculate 5 to the power of 2 and get 25.

$$\left(\frac{25}{3^{6}}\right)^{-\frac{1}{2}}\times \left(\frac{3^{4}\times 5^{3}}{3^{2}\times 5^{4}}\right)^{-1}$$

Calculate 3 to the power of 6 and get 729.

$$\left(\frac{25}{729}\right)^{-\frac{1}{2}}\times \left(\frac{3^{4}\times 5^{3}}{3^{2}\times 5^{4}}\right)^{-1}$$

Calculate \frac{25}{729} to the power of -\frac{1}{2} and get \frac{27}{5}.

$$\frac{27}{5}\times \left(\frac{3^{4}\times 5^{3}}{3^{2}\times 5^{4}}\right)^{-1}$$

Cancel out 3^{2}\times 5^{3} in both numerator and denominator.

$$\frac{27}{5}\times \left(\frac{3^{2}}{5}\right)^{-1}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{27}{5}\times \left(\frac{9}{5}\right)^{-1}$$

Calculate \frac{9}{5} to the power of -1 and get \frac{5}{9}.

$$\frac{27}{5}\times \frac{5}{9}$$

Multiply \frac{27}{5} and \frac{5}{9} to get 3.

$$3$$

Factor

$$3$$