Simplify $\sqrt x\left(\sqrt x+\frac1{\sqrt x}\right)$

Question

Would anyone be able to show me the answer (the question being to fully simplify this expression) and derivation? $$\sqrt x\left(\sqrt x+\frac1{\sqrt x}\right)$$

Answer 1

The expression is defined for $x>0$. You should use the following properties

1. $\sqrt{x}=x^{1/2}$
2. $x^ax^b=x^{a+b}$
3. $\dfrac1{x^a}=x^{-a}$
4. $x^0=1$

with $a=1/2$ in your case. \begin{align}\sqrt{x}\left(\sqrt{x}+\frac1{\sqrt{x}}\right)&=\sqrt{x}\sqrt{x}+\frac{\sqrt{x}}{\sqrt{x}}=x^{1/2}x^{1/2}+\frac{x^{1/2}}{x^{1/2}}\\[0.2cm]&=x^{1/2+1/2}+x^{1/2-1/2}=x^1+x^0\\[0.2cm]&=x+1\end{align}

Answer 2

Expanding it gives you

$ {\sqrt x}.{\sqrt x} + \left(\frac{\sqrt x .1}{\sqrt x}\right)$

$x+1$