How to simplify algebraic expressions with surds and brackets

Key concept: These questions look different, but the pattern is the same: remove brackets carefully, then gather the matching parts.

Step by step

Take it one line at a time

a) $3 + 2\sqrt{3} - 2 + 3\sqrt{3}$
Group the ordinary numbers and the surds:

$(3-2) + (2\sqrt{3}+3\sqrt{3}) = 1 + 5\sqrt{3}.$

b) $3 + 2\sqrt{3} - 4 + \sqrt{3}$

$(3-4) + (2\sqrt{3}+\sqrt{3}) = -1 + 3\sqrt{3}.$

c) $2\sqrt{6} + 3 - 2(1 + \sqrt{6})$
Expand the bracket first:

$2\sqrt{6}+3-2-2\sqrt{6}.$

Now the surd terms cancel:

$(2\sqrt{6}-2\sqrt{6}) + (3-2)=1.$

d) $3\sqrt{3} + 4 - (2\sqrt{3} - 1)$
Be careful with the minus sign:

$3\sqrt{3}+4-2\sqrt{3}+1.$

So

$(3\sqrt{3}-2\sqrt{3}) + (4+1)=\sqrt{3}+5.$

e) $2(\sqrt{5} + 2) - 2(\sqrt{5} - 2)$
Expand both brackets:

$2\sqrt{5}+4-2\sqrt{5}+4=8.$

f) $4(\sqrt{7} - \sqrt{2}) - (\sqrt{7} - \sqrt{2})$
Expand:

$4\sqrt{7}-4\sqrt{2}-\sqrt{7}+\sqrt{2}.$

Collect like terms:

$3\sqrt{7}-3\sqrt{2}=3(\sqrt{7}-\sqrt{2}).$

Final answers

• a) $1+5\sqrt{3}$
• b) $-1+3\sqrt{3}$
• c) $1$
• d) $5+\sqrt{3}$
• e) $8$
• f) $3(\sqrt{7}-\sqrt{2})$

Pitfall alert

The biggest mistake is dropping a minus sign when removing brackets, especially in expressions like $-(2\sqrt{3}-1)$. Also, don’t combine a constant with a surd term; $4+\sqrt{3}$ is already simplified.

Try different conditions

If the bracket had a positive sign in front, only the inside would be copied as-is. For example, $3\sqrt{3}+4+(2\sqrt{3}-1)=5\sqrt{3}+3$. If the coefficient changed, the expansion still works the same way: just distribute every outside factor to both terms inside the bracket.

Further reading

expanding brackets, collecting like terms, surd expressions