Comparing Expressions With Square Roots

Property When comparing expressions with radicals, simplify the radicals first. After that, perform any other operations and then compare the results. Crucially, $\sqrt{a} + \sqrt{b} \neq \sqrt{a+b}$.

Examples Compare $\sqrt{4} + \sqrt{36}$ and $\sqrt{9} + \sqrt{25}$. This simplifies to $2+6$ and $3+5$, which means $8=8$. Compare $\sqrt{9} + \sqrt{16}$ and $\sqrt{100}$. This simplifies to $3+4$ and $10$, which means $7 < 10$. Note that $\sqrt{16} + \sqrt{9} = 4 + 3 = 7$, but $\sqrt{16+9} = \sqrt{25} = 5$.

Explanation Don't get tricked! You have to simplify each square root individually before you can add or compare them. It's like opening two separate gift boxes; you can’t know which present is better until you see what’s actually inside each one of them first.