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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
up vote 1 down vote accepted

Here, we have the form $a^x\cdot b^y$, with the added knowledge that $y = \frac 12 x$.

So we can manipulate the expression to obtain the product of two bases raised to the same power, as you did, or as shown below. It all falls out from the laws of exponents, as they relate to real numbers.

$$\left(\dfrac 12\right)^x \cdot 4^{(x/2)} = \left(\dfrac 12\right)^x \cdot 2^{\left(2 (x/2)\right)} = \left(\dfrac 12\right)^x \cdot 2^{x}= \left(\dfrac 12\cdot 2\right)^x = 1^x = 1$$

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