How do you simplify (18+sqrt567)/9?

Jun 11, 2018

$2 + \sqrt{7}$

Explanation:

Begin by simplifying the radical, and making any squares possible inside the radical.

$\frac{18 + \sqrt{567}}{9}$ $\Rightarrow \frac{18 + \sqrt{{3}^{2} \cdot {3}^{2} \cdot 7}}{9}$

Now take the squares out and multiply their roots outside the radical:

$\frac{18 + \sqrt{{3}^{2} \cdot {3}^{2} \cdot 7}}{9} \Rightarrow$ $\frac{18 + \left(3 \cdot 3\right) \sqrt{7}}{9}$ $\Rightarrow \frac{18 + 9 \sqrt{7}}{9}$

Now simplify the top and bottom of the expression by canceling out $9$:

$\frac{\stackrel{2}{\cancel{18}} + \stackrel{1}{\cancel{9}} \sqrt{7}}{\cancel{9}} _ 1$ $\Rightarrow \frac{2 + 1 \sqrt{7}}{1} \Rightarrow 2 + \sqrt{7}$