Which of the following inequalities are true?
(A) \(\sqrt{11} + \sqrt{5} \lt 2\sqrt{8}\)
(B) \(\sqrt{2} \gt \sqrt[3]{3}\)
(C) \(\sqrt[5]{5} \lt \sqrt[3]{3}\)
(D) \(\sqrt{7} - \sqrt{6} \gt \sqrt{6} - \sqrt{5}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
Two positive numbers are compared by raising both to the same power, which keeps the order, until the roots disappear.
\((\sqrt{11} + \sqrt{5})^{2} = 16 + 2\sqrt{55}\) and \((2\sqrt{8})^{2} = 32 = 16 + 2\sqrt{64}\)
\((\sqrt{2})^{6} = 8\) and \((\sqrt[3]{3})^{6} = 9\)
\((\sqrt[5]{5})^{15} = 125\) and \((\sqrt[3]{3})^{15} = 243\)
For (D), each difference is rewritten by multiplying and dividing by the sum of the two roots:
\(\displaystyle \sqrt{7} - \sqrt{6} = \frac{1}{\sqrt{7} + \sqrt{6}}\)
\(\displaystyle \sqrt{6} - \sqrt{5} = \frac{1}{\sqrt{6} + \sqrt{5}}\)
So \(\sqrt{55} \lt \sqrt{64}\), \(8 \lt 9\), \(125 \lt 243\), and the first difference has the larger denominator.
Checking the statements
(A) \(\sqrt{11} + \sqrt{5} \lt 2\sqrt{8}\) — Correct. The squares are \(16 + 2\sqrt{55}\) and \(16 + 2\sqrt{64}\), and \(55 \lt 64\).
(B) \(\sqrt{2} \gt \sqrt[3]{3}\) — Incorrect. The sixth powers are 8 and 9, so \(\sqrt{2} \lt \sqrt[3]{3}\).
(C) \(\sqrt[5]{5} \lt \sqrt[3]{3}\) — Correct. The fifteenth powers are 125 and 243.
(D) \(\sqrt{7} - \sqrt{6} \gt \sqrt{6} - \sqrt{5}\) — Incorrect. \(\sqrt{7} - \sqrt{6}\) has the larger denominator \(\sqrt{7} + \sqrt{6}\), so it is the smaller difference.