Simplify \( \left(5^{1 / 2} \cdot 5^{1 / 4}\right) \) \( \left(5^{1 / 2} \cdot 5^{1 / 4}\right)=\square \)

To simplify \( \left(5^{1 / 2} \cdot 5^{1 / 4}\right) \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). So, applying that here: \[ 5^{1 / 2} \cdot 5^{1 / 4} = 5^{(1/2 + 1/4)} \] Next, we need a common denominator to add \( \frac{1}{2} \) and \( \frac{1}{4} \). The common denominator of 2 and 4 is 4. Thus, we can express \( \frac{1}{2} \) as \( \frac{2}{4} \): \[ \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \] Putting it all together, we get: \[ 5^{(1/2 + 1/4)} = 5^{3/4} \] Thus, \[ \left(5^{1 / 2} \cdot 5^{1 / 4}\right) = 5^{3 / 4} \] So, the final answer is: \[ \left(5^{1 / 2} \cdot 5^{1 / 4}\right) = 5^{3 / 4} \]