Adding and Subtracting Radicals
Remark 1 Recall the terminology we have for radical expressions:
index
↓
√
3
Coefficient −→ 7 41 ←− radicand
Definition Two radicals are called like radicals if they have exactly the
same index and exactly the same radicand, only the coefficients
can differ.
Remark 2 Like radicals are an extension of the idea of like terms.
ex.
−→
like terms:
10x3 and 45x3
↖
↗
similarly
←→
same variable to exactly
the same exponent
like√radicals: √
10 3 6 and 45 3 6
↖
↗
same radicand with exactly the same index
If either the radicands or the indexes differ, the radicals are NOT like
radicals.
√
√
ex.
−→ 10 3 6 and 10 4 6 are not like radicals because their indexes are different.
Remark 3 Sometimes two radicals with the same index can be changed into like
radicals even though they start with different radicands.
√
√
√
√
√
ex.
−→ 3 3 and 3 24 have different radicands, but 3 24 = 3 2 · 2 · 2 · 3 = 2 3 3.
Therefore, before we can judge if two radicals are like radicals, we must
simplify.
Main Idea • Only like radicals can be added or subtracted.
• To add or subtract like radicals, collect the coefficients (but
do not change the radical).
√
√
√
ex.
4
4
−→ 5 13+ 7 13 = 12 4 13
Example 1 Simplify. Assume all variables represent positive values.
√
√
√
√
√
a 7 54 − 3 24 − 6
b 75x2 + 12x2
√
3
√
3
c 5x 81x + 6x 24x
√
e
√
√
27 3 3
3
−
+√
2
2
4
√
√
d 9 44 − 10 13