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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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    $\begingroup$ Check: $\sqrt{8^2+(8\sqrt3)^2} = 8\sqrt{1+3}$ $\endgroup$
    – Macavity
    Commented Oct 20, 2015 at 8:16
  • $\begingroup$ When you "took out" the $8^2$ the sum in the root went missing. Where did it go? $\endgroup$
    – Christoph
    Commented Oct 20, 2015 at 8:17

2 Answers 2

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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}

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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}

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