Simplifying $\frac{1}{2\cdot\sqrt{e^x}}\cdot e^x$

Question

We have to simplify:

$$\frac{1}{2\cdot\sqrt{e^x}}\cdot e^x$$

I came to the conclusion that the answer would be:

$$\frac{e^x}{2\cdot\sqrt{e^x}}$$

But I was wrong and it was: $$\frac{\sqrt{e^x}}{2}$$

Where did I go wrong and why?

Thanks

Answer 1

You did not simplify the question.

notice that $$\frac12\frac{e^x}{\sqrt{e^x}}=\frac12\frac{e^x}{e^\frac{x}{2}}=\frac12\exp(x-\frac{x}2)$$

Answer 2

$$\frac{1}{2\sqrt{e^x}}\cdot e^x=\frac{1}{2\sqrt{e^x}}\cdot (\sqrt{e^x}\sqrt{e^x})=\frac{\require{cancel}\cancel{\sqrt{e^x}}}{2\cancel{\sqrt{e^x}}}\cdot\sqrt{e^x}=\frac{\sqrt{e^x}}2$$

Answer 3

$$\require{cancel}\frac{1}{2\cdot\sqrt{e^x}}\cdot e^x = \frac{1}{2\cdot\sqrt{e^x}}\cdot e^x\cdot\frac{\sqrt{e^x}}{\sqrt{e^x}} = \frac{1}{2}\cdot\sqrt{e^x}\cdot\frac{e^x}{\sqrt{e^x}\cdot\sqrt{e^x}} = \frac{1}{2}\cdot\sqrt{e^x}\cdot\frac{\cancel{e^x}}{\cancel{e^x}} = \frac{1}{2}\cdot\sqrt{e^x}$$

Answer 4

For $a>0$:

$$\frac{a}{\sqrt{a}}=\frac{(\sqrt{a})^2}{\sqrt{a}}=\sqrt{a}.$$