How to Simplify Square Roots, Step by Step
A calculator will tell you that √72 is about 8.485, but most algebra homework wants the exact answer, 6√2. Getting there is called simplifying the square root, and it uses one idea over and over: pull every perfect square out from under the radical sign.
The Common Core grade 8 standards start this topic by asking students to evaluate square roots of small perfect squares and to "know that √2 is irrational." In high school, the standards ask students to "rewrite expressions involving radicals and rational exponents using the properties of exponents." The steps in between follow.
1. The perfect squares you need
Simplifying is much faster if you recognize these on sight:
| n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| n² | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
2. The rule behind every step
For numbers that are not negative, the square root of a product is the product of the square roots:
√(ab) = √a · √b
So if you can write the number under the radical as a perfect square times something else, the perfect square comes out as a whole number.
3. Example 1: √72
- Find the largest perfect square that divides 72. Checking from the top of the table: 64 no, 49 no, 36 yes (72 = 36 · 2).
- Split: √72 = √36 · √2
- Simplify: √72 = 6√2
Decimal check: √2 is about 1.41421, and 6 · 1.41421 = 8.485, which matches √72. Checking with a calculator like this confirms you did not lose a factor.
If you pick a smaller square first, you are not wrong, just not finished. Using 9: √72 = √9 · √8 = 3√8. But 8 still contains the perfect square 4, so 3√8 = 3 · 2√2 = 6√2. A square root is fully simplified only when the number left inside has no perfect square factor other than 1.
4. More quick examples
| Radical | Largest perfect square factor | Simplified |
|---|---|---|
| √50 | 25 (50 = 25 · 2) | 5√2 |
| √48 | 16 (48 = 16 · 3) | 4√3 |
| √45 | 9 (45 = 9 · 5) | 3√5 |
| √98 | 49 (98 = 49 · 2) | 7√2 |
5. Adding and subtracting radicals
You can only combine radicals that have the same number under the sign, the same way you combine like terms such as 3x + 5x.
- 3√2 + 5√2 = 8√2
- 3√2 + 5√3 cannot be combined
Sometimes radicals only look unlike. Simplify each one first:
√8 + √18 = 2√2 + 3√2 = 5√2
Decimal check: √8 + √18 is about 2.828 + 4.243 = 7.071, and 5√2 is also about 7.071.
6. The most common mistake
The product rule does not work for sums.
√(36 + 64) is not √36 + √64.
- Correct: √(36 + 64) = √100 = 10
- Wrong: √36 + √64 = 6 + 8 = 14
Always add what is inside the radical first, then take the root.
7. Multiplying radicals
Multiply the numbers inside, then simplify the result:
√6 · √15 = √90 = √(9 · 10) = 3√10
A related fact worth memorizing: √a · √a = a. For example, √7 · √7 = 7.
8. Rationalizing the denominator
Many teachers want no radical in the denominator of a final answer. To remove one, multiply the top and bottom by that radical:
6 / √3 = (6 · √3) / (√3 · √3) = 6√3 / 3 = 2√3
Multiplying by √3/√3 is multiplying by 1, so the value does not change. Decimal check: 6 / √3 is about 3.464, and 2√3 is also about 3.464.
9. Cube roots work the same way
For a cube root, pull out perfect cubes (8, 27, 64, 125) instead of perfect squares:
∛54 = ∛(27 · 2) = 3∛2
10. Connecting radicals to exponents
The high school standards explain that radicals can be written with fractional exponents, and that the meaning comes from extending the exponent rules. Their example: 5^(1/3) is defined as the cube root of 5 "because we want (5^(1/3))³ = 5^((1/3)3) to hold, so (5^(1/3))³ must equal 5."
In the same way, √x can be written as x^(1/2). This is why simplifying √72 and rewriting 72^(1/2) are the same problem in two notations.
11. Why some answers stay as radicals
The grade 8 standards point out that √2 is irrational: its decimal never ends and never repeats. Any decimal you write for it is a rounded approximation. The simplified radical 6√2 is exact, which is why algebra classes prefer it.
12. Key takeaways
- Find the largest perfect square factor, split the radical, and take that square root out.
- A radical is fully simplified when the number inside has no perfect square factor other than 1.
- Combine only like radicals, and simplify first to reveal hidden like terms.
- The square root of a sum is not the sum of the square roots.
- Rationalize a denominator by multiplying the top and bottom by the radical, and check every answer with a quick decimal comparison.