Which one of the following is true? (a) √5 + √3 > √6 + √2 (b) √5 + √3 < √6 + √2 (c) √5 + √3 = √6 + √2 (d) (√5 + √3)(√6 + √2) = 1

Solution

To determine which of the given statements is true, we evaluate each option:

Option (a): \(\sqrt{5}+\sqrt{3}>\sqrt{6}+\sqrt{2}\)

We calculate both sides of the inequality:

• Left side: \(\sqrt{5} + \sqrt{3}\)
• Right side: \(\sqrt{6} + \sqrt{2}\)

Approximating the square roots:

• \(\sqrt{5} \approx 2.236\)
• \(\sqrt{3} \approx 1.732\)
• \(\sqrt{6} \approx 2.449\)
• \(\sqrt{2} \approx 1.414\)

Summing up the approximations:

• Left side: \(2.236 + 1.732 = 3.968\)
• Right side: \(2.449 + 1.414 = 3.863\)

Since \(3.968 > 3.863\), option (a) \(\sqrt{5}+\sqrt{3}>\sqrt{6}+\sqrt{2}\) is true.

Option (b): \(\sqrt{5}+\sqrt{3}<\sqrt{6}+\sqrt{2}\)

From our calculation above, we know that the left side is greater than the right side, making this option false.

Option (c): \(\sqrt{5}+\sqrt{3}=\sqrt{6}+\sqrt{2}\)

As shown, the two sides are not equal, making this option false.

Option (d): \((\sqrt{5}+\sqrt{3})(\sqrt{6}+\sqrt{2})=1\)

This option can be quickly dismissed without calculation, as the product of these sums, given their approximate values, clearly does not equal 1.

Thus, the correct answer is (a) \(\sqrt{5}+\sqrt{3}>\sqrt{6}+\sqrt{2}\).