Which of the following is correct? (i) \(\sqrt{7} + \sqrt{2} > \sqrt{6} + \sqrt{3}\)
(ii) \(\sqrt{7} + \sqrt{2} < \sqrt{6} + \sqrt{3}\)
(iii) \(\sqrt{7} + \sqrt{2} = \sqrt{6} + \sqrt{3}\)
(ii)
Step 1 – square both expressions to compare:
\((\sqrt{7} + \sqrt{2})^2 = 7 + 2 + 2\sqrt{14} = 9 + 2\sqrt{14}\)
\((\sqrt{6} + \sqrt{3})^2 = 6 + 3 + 2\sqrt{18} = 9 + 2\sqrt{18}\)
Step 2 – compare:
Since \(\sqrt{18} > \sqrt{14}\) (because \(18 > 14\)), we get \(9 + 2\sqrt{18} > 9 + 2\sqrt{14}\).
Both expressions are positive, so taking square roots preserves the inequality:
\(\sqrt{6} + \sqrt{3} > \sqrt{7} + \sqrt{2}\), i.e. \(\sqrt{7} + \sqrt{2} < \sqrt{6} + \sqrt{3}\).
Hence the correct option is (ii).
Similar Questions
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$\frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}}$ - The value of $\sqrt{11 - 2\sqrt{30}} - \frac{1}{\sqrt{11 - 2\sqrt{30}}}$ is:
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Rationalize:
$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}$- Simplify:
$\sqrt{3}(\sqrt{12} - \sqrt{27})$ - Simplify:
$\frac{\sqrt{5+2\sqrt{6}} + \sqrt{5-2\sqrt{6}}}{\sqrt{2}}$
Important Questions from Surds and Indices
The value of \(\frac{{{{\left( {251} \right)}^3} + {{\left( {249} \right)}^3}}}{{25.1 \times 25.1 - 624.99 + 24.9 \times 24.9}}\) is 5 × 10 k , where the value of k is :
Find the value of m in \(\left(\frac{2}{7}\right)^{-3} \times \left(\frac{2}{7}\right)^{-5}=\left (\frac{2}{7}\right)^{-3m+1}\)
If √625 = 25; then√(.00000625/25)is:
A. 0.0025
B. 0.001
C. 0.0001
D. 0.0005Find the value of:
\(\sqrt{150}-\sqrt{54}-\sqrt{24}\)
If \(\sqrt{4624}=68\) , then the value of:
\(\sqrt{46.24}+\sqrt{0.4624}+\sqrt{0.004624}\)
