The first step is take out $8^2$, then $8\sqrt{ (8\sqrt{3})^2 }$.
What are the correct next steps?
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1$\begingroup$ Check: $\sqrt{8^2+(8\sqrt3)^2} = 8\sqrt{1+3}$ $\endgroup$– MacavityOct 20, 2015 at 8:16
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$\begingroup$ When you "took out" the $8^2$ the sum in the root went missing. Where did it go? $\endgroup$– ChristophOct 20, 2015 at 8:17
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2 Answers
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\begin{equation} \begin{split} \text{n}&:=\sqrt{8^2+(8\sqrt{3})^2}\\ \\ &=\sqrt{64+(8\sqrt{3})^2}\\ \\ &=\sqrt{64+8^2\cdot\left(\sqrt{3}\right)^2}\\ \\ &=\sqrt{64+64\cdot 3}\\ \\ &=\sqrt{64+192}\\ \\ &=\sqrt{256}\\ \\ &=\sqrt{16^2}\\ \\ &=16 \end{split}\tag1 \end{equation}
answered Oct 20, 2015 at 8:11
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Alternative to Jan Eerland his answer:
\begin{align} \sqrt{8^2 + (8\sqrt{3})^2} = \sqrt{8^2 + 8^2 (\sqrt{3})^2} = \sqrt{8^2 + 8^2 \cdot 3} = \sqrt{8^2(1 + 3)} = \sqrt{8^2 \cdot 2^2} = 8 \cdot 2 = 16. \end{align}
