Show why the square root addition and subtraction rules are false

Expert intro: A rule can look believable and still be wrong. With radicals, one clean counterexample is enough to show that a statement is not always true.

Detailed walkthrough

To prove a rule is incorrect, it is enough to find one counterexample.

a) Show that $\sqrt{m} + \sqrt{n} = \sqrt{m+n}$ is incorrect

Choose positive integers $m=1$ and $n=1$.

Left side:

$\sqrt{1} + \sqrt{1} = 1 + 1 = 2$

Right side:

$\sqrt{1+1} = \sqrt{2}$

Since $2 \ne \sqrt{2}$, the rule is false.

b) Show that $\sqrt{m} - \sqrt{n} = \sqrt{m-n}$ is incorrect

Choose positive integers $m=9$ and $n=4$.

Left side:

$\sqrt{9} - \sqrt{4} = 3 - 2 = 1$

Right side:

$\sqrt{9-4} = \sqrt{5}$

Since $1 \ne \sqrt{5}$, the rule is false.

A compact way to remember it: square roots do not distribute over addition or subtraction.

💡 Pitfall guide

The mistake is treating the radical sign like ordinary arithmetic parentheses. It does not “pass through” addition or subtraction. If you want to test a rule, use small perfect squares first; they make counterexamples very obvious.

🔄 Real-world variant

If the numbers under the root are not perfect squares, the rule still fails. For example, $\sqrt{2}+\sqrt{3}\neq\sqrt{5}$. The only time expressions like this can simplify is when the radicals are like terms after simplification, not because the radicands were added or subtracted.

🔍 Related terms

counterexample, radical rules, square root properties