# Simplifying exponentials of the form $\,a^x \cdot b^y$

I am given the exponential $\left(\dfrac{1}{2}\right)^x\cdot 4^{(x/2)}$. While my intuition screams that this can be simplified to $\dfrac{2^x}{2^x} = 1$, I am unable to see a concrete mathematical explanation for this.

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$4^y = (2^2)^y = 2^{2y}$, and $(1/2)^z = (2^{-1})^z = 2^{-z}$ –  Prahlad Vaidyanathan Sep 22 '13 at 17:50

Here, we have the form $a^x\cdot b^y$, with the added knowledge that $y = \frac 12 x$.
$$\left(\dfrac 12\right)^x \cdot 4^{(x/2)} = \left(\dfrac 12\right)^x \cdot 2^{\left(2 (x/2)\right)} = \left(\dfrac 12\right)^x \cdot 2^{x}= \left(\dfrac 12\cdot 2\right)^x = 1^x = 1$$