Aus Online Mathematik Brückenkurs 1

That which is under the root sign is the same as \displaystyle (-3)^{2} = 9 and because \displaystyle 9 = 3\cdot 3 = 3^{2}, hence

\displaystyle \sqrt{(-3)^{2}} = \sqrt{9} = 9^{1/2} = \bigl(3^{2}\bigr)^{1/2} = 3^{2\cdot\frac{1}{2}} = 3^{1} = 3.

Note: The calculation \displaystyle \sqrt{(-3)^{2}} = \bigl((-3)^{2}\bigr)^{1/2} = (-3)^{2\cdot \frac{1}{2}} = (-3)^1 = -3 is wrong at the second equals sign. Remember that the power rules apply when the base is positive.