Algebra 1 · Unit 6
Combine like square roots and cube roots. Multiply by putting the insides together, then simplify.
Jump to section:
Like radicals
You can add or subtract only matching leftovers: 3√2 + 5√2 = 8√2. Different leftovers stay separate.
Simplify first
Often two radicals match only after you pull out a perfect square or cube.
Multiply
√a × √b = √(ab). Cube roots multiply the same way.
Do not add insides
√a + √b is not √(a + b).
Like radicals, or multiply two square roots.
Like leftovers, so add the coefficients.
5√5
Answer: 5√5
4√2
Answer: 4√2
√36 = 6
Answer: 6
7∛4
Answer: 7∛4
√16 = 4
Answer: 4
5√3
Answer: 5√3
Simplify a radical first so the leftovers match.
√8 = 2√2
√18 = 3√2
2√2 + 3√2 = 5√2
Answer: 5√2
√12 = 2√3
2√3 + √3 = 3√3
Answer: 3√3
2 × 5√2 − 2√2
10√2 − 2√2 = 8√2
Answer: 8√2
2∛2 + 3∛2 = 5∛2
Answer: 5∛2
6 × 7 = 42
Answer: 42
2√5 − √5 = √5
Answer: √5
Mixed leftovers, cube roots, and a product that still needs simplifying.
√48 = 4√3
√75 = 5√3
√12 = 2√3
4 + 5 − 2 = 7, so 7√3
Answer: 7√3
2 × 3√2 × √2 = 6 × 2 = 12
Answer: 12
∛24 = 2∛3
∛81 = 3∛3
5∛3
Answer: 5∛3
√144 = 12
Answer: 12
4 × 2√3 − 2 × 3√3
8√3 − 6√3 = 2√3
Answer: 2√3
−3∛2 − (−2∛2)
−3∛2 + 2∛2 = −1∛2
Answer: −1∛2
√8 + √2 = 2√2 + √2 = 3√2
3√2 × 3√2 = 18, so 3√2 = √18
√18 > √10 because 18 > 10.
Answer: 3√2, which is larger than √10
a × 2√3 + √3 = 7√3
(2a + 1)√3 = 7√3
2a + 1 = 7
a = 3
Answer: a = 3
∛8 = 2 and ∛27 = 3, so the sum is 5.
You add cube roots only when the leftovers match. 8 and 27 are already perfect cubes, so you add the integers.
Answer: 5
Need help with radical operations?
Book a Session with Amy