To simplify a square root, find the largest perfect square that divides the number under the root, split it off, and take its square root. For example, 72=36⋅2=62. You can only add or subtract radicals with the same number under the root, like 32+52=82. A simplified answer has no square root left in a denominator.
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The key idea
A square root of a product splits into a product of square roots. If one of the pieces is a perfect square, its root is a whole number and can leave the radical sign:
ab=a⋅bso72=36⋅2=62
It helps to know the perfect squares by sight: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. A radical is fully simplified when no perfect square (other than 1) divides the number under the root, and there is no root in a denominator.
Worked examples
Example 1: pull out the largest perfect square
Problem Simplify 72.
The largest perfect square that divides 72 is 36.
72=36⋅2
Split the root and take 36=6.
36⋅2=36⋅2=62
Answer62
Example 2: a radical with a variable
Problem Simplify 50x3, where x≥0.
Split into the perfect-square part and the rest. 50=25⋅2 and x3=x2⋅x.
50x3=25x2⋅2x
25x2=5x, because (5x)2=25x2. The 2x stays inside.
25x2⋅2x=5x2x
Answer5x2x
Example 3: add and subtract radicals
Problem Simplify 312+27−75.
These look unlike, so simplify each one first. 12=4⋅3, 27=9⋅3, 75=25⋅3.
312+27−75=3⋅23+33−53
Now all three are multiples of 3. Combine them like 6y+3y−5y.
63+33−53=43
Answer43
Example 4 (test-hard): rationalize, then combine
Problem Simplify 510+20.
Rationalize: multiply the top and the bottom by 5. That is multiplying by 1, so the value does not change, and 5⋅5=5.
510=510⋅55=5105=25
Simplify the second radical.
20=4⋅5=25
Now they match, so add.
25+25=45
Answer45
Common mistakes
Splitting a root over a sum.9+16=25=5, not 3+4=7. Splitting only works for multiplication.
Stopping too early.72=218 is true but not finished, since 18 still contains 9. Fix: use the largest perfect square, or keep going.
Adding unlike radicals.2+3 is not 5. Only radicals with the same number inside combine.
Losing the outside number. In 312, the 2 that comes out multiplies the 3: 3⋅23=63.
Leaving a root in the denominator. Most teachers want 510 written as 25.
Quick methods
Practice
5 practice questions
Simplify 48.
43
34
62
163
Show answer
Answer: 43
48=16⋅3, and 16=4. The choice 163 forgets to take the square root of 16. 62 is 72, and 34 is just 6.
Simplify 218+8.
82
226
326
102
Show answer
Answer: 82
18=32, so 218=62. 8=22. Together that is 82. The choices with 26 add the numbers under the roots, which is not allowed.
Simplify (32)(46).
243
123
712
128
Show answer
Answer: 243
Multiply outside by outside and inside by inside: 1212. Then 12=23, so the result is 243. The choice 123 forgets the 2 that came out of 12.
Rationalize the denominator: 28.
42
82
4
22
Show answer
Answer: 42
Multiply top and bottom by 2: 282=42. The choice 82 multiplies the top but forgets that the bottom became 2.
What is 81+144?
15
21
12.5
225
Show answer
Answer: 15
Add first: 81+144=225, and 225=15. The trap 21 is 9+12, which splits the root over a sum. 225 forgets to take the root.
Frequently asked questions
What makes a square root fully simplified?
Three things. No perfect square other than 1 divides the number under the root. There is no fraction under the root. And there is no root in the denominator. So 62 is simplified, but 218 and 510 are not.
Can you add square roots?
Only like ones, with the same number under the root. Treat 3 like a variable: 63+33=93. If they look different, simplify each first, because 12 and 27 both turn into multiples of 3.
Why do we rationalize the denominator?
It is a convention that makes answers easy to compare and combine. 510 and 25 are the same number, but the second form can be added to other multiples of 5 right away, as in Example 4.
How do I simplify a square root with variables?
Split each variable power into an even part and what is left. Even powers come out at half the exponent: x2=x and x6=x3 when x is not negative. So x5=x4⋅x=x2x.
