Simplify. Assume all variables represent positive real numbers.
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Question

Quizplus · Accepted Answer

We can simplify the expression using the fact that $x^{a+b}=x^ax^b$ and $(x^a)^b=x^{ab}$. Thus,

\begin{align*}
\sqrt{2}^{2x+1}\cdot (\sqrt{2}^{-x}+2\sqrt{2}^x) &= (\sqrt{2}^2)^{x} \sqrt{2} \cdot \left(\frac{1}{\sqrt{2}}ight)^x +(\sqrt{2}^2)^{x}\cdot 2 \cdot \sqrt{2}^x \
&= 2^x \sqrt{2} \cdot 2^{-x} + 2 \cdot 2^x \sqrt{2}^x \
&= \sqrt{2} + 2\sqrt{2}^x \
&= \sqrt{2} \cdot (1+2^x) \
&= \sqrt{2}^{1} \cdot 2^{x} \
&= \sqrt{2}^{2\cdot \frac{x}{2}+1}\cdot 2^{0} \
&= \sqrt{2}^{2x+1}\cdot 1.
\end{align*}

Therefore, the answer is $\boxed{	extbf{(C) }11ebb63d_1bff_2bd3_b851_2150a38bcd9e_TB8475_11}$.

Simplify. Assume All Variables Represent Positive Real Numbers