(a) Simplify : \(3\sqrt{75} - \sqrt{12} + \sqrt{108}\), leaving the answer in surd form (radicals). (b) If \(124_{n} = 232_{five}\), find n.

Answer Details

(a) Simplify \(3\sqrt{75} - \sqrt{12} + \sqrt{108}\)

Reduce each surd to lowest terms by taking out perfect squares:

\[\sqrt{75} = \sqrt{25\times 3} = 5\sqrt{3} \;\Rightarrow\; 3\sqrt{75} = 15\sqrt{3}\]

\[\sqrt{12} = \sqrt{4\times 3} = 2\sqrt{3}\]

\[\sqrt{108} = \sqrt{36\times 3} = 6\sqrt{3}\]

Combine the like surds:

\[15\sqrt{3} - 2\sqrt{3} + 6\sqrt{3} = 19\sqrt{3}\]

(b) If \(124_{n} = 232_{five}\), find \(n\).

Convert the right side to base ten:

\[232_{five} = 2(5^{2}) + 3(5) + 2 = 50 + 15 + 2 = 67\]

Express the left side in base ten:

\[124_{n} = 1(n^{2}) + 2(n) + 4 = n^{2} + 2n + 4\]

Set them equal and solve:

\[n^{2} + 2n + 4 = 67 \;\Rightarrow\; n^{2} + 2n - 63 = 0\]

\[(n+9)(n-7) = 0\]

Since a base must be positive, \(n = 7\).