\(\sqrt{170 - 20 \sqrt{30}} = \sqrt{a} - \sqrt{b}\)
Squaring both sides,
\(170 - 20\sqrt{30} = a + b - 2\sqrt{ab}\)
Equating the rational and irrational parts, we have
\(a + b = 170 ... (1)\)
\(2 \sqrt{ab} = 20 \sqrt{30}\)
\(2 \sqrt{ab} = 2 \sqrt{30 \times 100} = 2 \sqrt{3000} \)
\(ab = 3000 ... (2)\)
From (2), \(b = \frac{3000}{a}\)
\(a + \frac{3000}{a} = 170 \implies a^{2} + 3000 = 170a\)
\(a^{2} - 170a + 3000 = 0\)
\(a^{2} - 20a - 150a + 3000 = 0\)
\(a(a - 20) - 150(a - 20) = 0\)
\(\text{a = 20 or a = 150}\)
\(\therefore b = \frac{3000}{20} = 150 ; b = \frac{3000}{150} = 20\)
\(\sqrt{170 - 20\sqrt{30}} = \sqrt{20} - \sqrt{150}\) or \(\sqrt{150} - \sqrt{20}\)
= \(2\sqrt{5} - 5\sqrt{6}\) or \(5\sqrt{6} - 2\sqrt{5}\)
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