How to simplify surds by collecting like terms

Key takeaway: When surds share the same root, you can only combine the coefficients after first simplifying each radical. That small order matters a lot here.

Work each expression into a simpler radical first

a) $a\sqrt{20} + \sqrt{5}$
$\sqrt{20} = \sqrt{4\cdot 5} = 2\sqrt{5}$, so

$a\sqrt{20}+\sqrt{5}=a(2\sqrt{5})+\sqrt{5}=(2a+1)\sqrt{5}.$

b) $\sqrt{12} + \sqrt{27}$

$\sqrt{12}=2\sqrt{3}, \quad \sqrt{27}=3\sqrt{3}$

so

$\sqrt{12}+\sqrt{27}=2\sqrt{3}+3\sqrt{3}=5\sqrt{3}.$

c) $4\sqrt{3} - 2\sqrt{27}$

$\sqrt{27}=3\sqrt{3}$

therefore

$4\sqrt{3}-2\sqrt{27}=4\sqrt{3}-2(3\sqrt{3})=4\sqrt{3}-6\sqrt{3}=-2\sqrt{3}.$

d) $3\sqrt{8} + 2\sqrt{18}$

$\sqrt{8}=2\sqrt{2}, \quad \sqrt{18}=3\sqrt{2}$

so

$3\sqrt{8}+2\sqrt{18}=3(2\sqrt{2})+2(3\sqrt{2})=6\sqrt{2}+6\sqrt{2}=12\sqrt{2}.$

e) $\sqrt{75} - 2\sqrt{48}$

$\sqrt{75}=5\sqrt{3}, \quad \sqrt{48}=4\sqrt{3}$

thus

$\sqrt{75}-2\sqrt{48}=5\sqrt{3}-2(4\sqrt{3})=5\sqrt{3}-8\sqrt{3}=-3\sqrt{3}.$

f) $3\sqrt{27} + 2\sqrt{12}$

$\sqrt{27}=3\sqrt{3}, \quad \sqrt{12}=2\sqrt{3}$

so

$3\sqrt{27}+2\sqrt{12}=3(3\sqrt{3})+2(2\sqrt{3})=9\sqrt{3}+4\sqrt{3}=13\sqrt{3}.$

Final answers

• a) $(2a+1)\sqrt{5}$
• b) $5\sqrt{3}$
• c) $-2\sqrt{3}$
• d) $12\sqrt{2}$
• e) $-3\sqrt{3}$
• f) $13\sqrt{3}$

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Pitfalls the pros know 👇 A common slip is trying to combine terms before simplifying the radicals. For example, $\sqrt{12}+\sqrt{27}$ is not $\sqrt{39}$. Also watch signs carefully: in part (c) and (e), the minus sign must be distributed after simplification.

What if the problem changes? If the coefficients were different, the method stays the same: simplify each surd first, then collect like radicals. For instance, $5\sqrt{12}+\sqrt{27}=10\sqrt{3}+3\sqrt{3}=13\sqrt{3}$. If the roots are unlike, such as $\sqrt{2}+\sqrt{3}$, they cannot be combined any further.

Tags: surd simplification, like terms, square roots