Evaluate $\left(2^{\left(1\frac{1}{4}\right)}\right)^{\frac{2}{5}} \cdot \left(4^{\left(3\frac{1}{8}\right)}\right)^{\frac{2}{25}}$.
This problem involves exponents and fractions, so I want to simplify them as much as possible. One way to do that is to write everything as powers of the same base, since that makes it easier to apply the exponent rules. I notice that both 2 and 4 are powers of 2, so I can rewrite them as such. $2^{\left(1\frac{1}{4}\right)} = 2^{\left(\frac{5}{4}\right)}$ and $4^{\left(3\frac{1}{8}\right)} = (2^2)^{\left(\frac{25}{8}\right)} = 2^{\left(2 \cdot \frac{25}{8}\right)} = 2^{\left(\frac{50}{8}\right)}$. Now I can rewrite the original expression as $\left(2^{\left(\frac{5}{4}\right)}\right)^{\frac{2}{5}} \cdot \left(2^{\left(\frac{50}{8}\right)}\right)^{\frac{2}{25}}$. Next, I want to apply the rule that says $(a^b)^c = a^{bc}$, which means I can multiply the exponents inside and outside the parentheses. That gives me $2^{\left(\frac{5}{4} \cdot \frac{2}{5}\right)} \cdot 2^{\left(\frac{50}{8} \cdot \frac{2}{25}\right)}$. Now I can simplify the fractions by canceling out common factors. $\frac{5}{4} \cdot \frac{2}{5} = \frac{2}{4} = \frac{1}{2}$ and $\frac{50}{8} \cdot \frac{2}{25} = \frac{4}{8} = \frac{1}{2}$. So the expression becomes $2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}$. Finally, I can apply the rule that says $a^b \cdot a^c = a^{b+c}$, which means I can add the exponents of the same base. That gives me $2^{\left(\frac{1}{2} + \frac{1}{2}\right)} = 2^1 = 2$.

