Same Math › Algebra 1 · grades 8–10 › Simplifying radicals
How do you simplify a square root?
To simplify a square root, split the number into a perfect square times what's left, then take the square root of the perfect square. √72 = √36 · √2 = 6√2. A radical is in simplest form when nothing under the root has a perfect-square factor. Kids then add like radicals (2√3 + 5√3 = 7√3) and multiply them (√2 · √8 = √16 = 4).
- Find the biggest perfect square that divides the number.
- Split the root into two roots.
- The perfect square comes out; the rest stays under the root.
Why schools teach it
Exact answers instead of decimals
Students meet square roots in eighth grade (8.EE.A.2) and rewrite radical expressions in high school (HSN.RN.A.2). Geometry depends on it: the diagonal of a square with side 5 is 5√2, and teachers want that, not 7.07.
Knowing the perfect squares to 144 by heart makes this fast. Kids who use a factor tree instead pull out pairs: a pair of 3s under the root comes out as one 3.
At the kitchen table
What to say, and what to skip
Say this
Which perfect squares go into this number?
Start big: 100, 81, 64, 49, 36, 25, 16, 9, 4.Is there still a perfect square left under the root?
Then keep going.To add radicals, are the parts under the roots the same?
Like terms only.When you multiply roots, what can you do first?
Multiply under one root, then simplify.
Skip this
- Leaving a perfect square behind.√72 = 2√18 isn't done; 18 still has a 9.
- Pulling out the square instead of its root.√36 comes out as 6, not 36.
- Adding unlike radicals.√2 + √3 is not √5. Leave it.
See it in action
Simplifying 3√50
Find the biggest perfect square in 50
50 = 25 · 2Take its root out front
3 · 5√2Multiply the outside numbers
15√2
Check: 15² · 2 = 450 and 3² · 50 = 450 ✓.
Free practice sheet
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Parents also ask
What if there's no perfect square factor?
Then it's already simplified, like √15 or √30.
What about a fraction under the root?
Split it: √(9/4) = 3/2. A root left on the bottom gets rationalized, which is usually taught next.
What about cube roots?
Same idea with perfect cubes: ∛54 = ∛27 · ∛2 = 3∛2.
Do they ever just use a calculator?
For a decimal answer, yes. When the directions say exact or simplest radical form, no.