Fix mistakes when simplifying radical expressions

Key concept: A lot of radical mistakes come from trying to combine unlike terms or from simplifying a square root too aggressively. The good news is that the rules are pretty strict, which makes them easy to check.

Step by step

a) What mistakes did the student make?

Expression 1:

$5\sqrt{3} + 3\sqrt{2} + 6\sqrt{3}$

The student wrote $11\sqrt{6} + 3\sqrt{2}$, which is wrong because:

• they combined $5\sqrt{3}$ and $6\sqrt{3}$ correctly in spirit, but not in form;
• you can add only like radicals, so $5\sqrt{3} + 6\sqrt{3} = 11\sqrt{3}$, not $11\sqrt{6}$;
• the radicand does not get multiplied when adding like terms.

Expression 2:

$2\sqrt{5} + \sqrt{18} + \sqrt{20} - 2\sqrt{3}$

The student’s $\sqrt{2} + \sqrt{38}$ is incorrect because they simplified and combined terms that are not like radicals.

First rewrite the roots:

$\sqrt{18} = 3\sqrt{2}, \quad \sqrt{20} = 2\sqrt{5}$

Then combine like terms only.

b) Correct answers

1.

$5\sqrt{3} + 3\sqrt{2} + 6\sqrt{3} = 11\sqrt{3} + 3\sqrt{2}$

2.

= 2\sqrt{5} + 3\sqrt{2} + 2\sqrt{5} - 2\sqrt{3} = 4\sqrt{5} + 3\sqrt{2} - 2\sqrt{3}$$ So the corrected answers are: - $11\sqrt{3} + 3\sqrt{2}$ - $4\sqrt{5} + 3\sqrt{2} - 2\sqrt{3}$ ### Pitfall alert The biggest trap is changing the radicand when you add radicals. $a\sqrt{m} + b\sqrt{m} = (a+b)\sqrt{m}$, but $a\sqrt{m} + b\sqrt{n}$ cannot be combined unless the radicals match exactly. Also, simplify each square root first before combining. ### Try different conditions If the expression includes a coefficient outside the radical, simplify the square root first and then combine like terms. For example, $4\sqrt{8} = 8\sqrt{2}$, which may then combine with other $\sqrt{2}$ terms. If there are variables inside the radicals, the same rule still applies: only like radicals can be added or subtracted. ### Further reading like radicals, simplify square roots, rational exponents