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Problem #46 · September 11, 2026

Powers and roots

Which of the following relations is true?
  1. A6+8<27\sqrt{6}+\sqrt{8} < 2\sqrt{7}
  2. B6+8>27\sqrt{6}+\sqrt{8} > 2\sqrt{7}
  3. C6+8=27\sqrt{6}+\sqrt{8} = 2\sqrt{7}
  4. D6+8=14\sqrt{6}+\sqrt{8} = \sqrt{14}
  5. E6+8=48\sqrt{6}+\sqrt{8} = \sqrt{48}

WORKED EXAMPLE

Solution

Both sides are positive, so we compare their squares: (6+8)2=14+248(\sqrt{6}+\sqrt{8})^{2} = 14 + 2\sqrt{48} and (27)2=28=14+249(2\sqrt{7})^{2} = 28 = 14 + 2\sqrt{49}. Since 48<49\sqrt{48} < \sqrt{49}, we have 6+8<27\sqrt{6}+\sqrt{8} < 2\sqrt{7}.
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