What is a square root?
If n × n = m, then n is a square root of m. Every positive number has two square roots, one positive and one negative: 13 × 13 and −13 × −13 are both 169. The √ sign means the positive one, so √169 = 13. The square root of a negative number is not a real number, because no number times itself is negative, but it can be written with the imaginary number i, which is √−1: √−49 = √49 × √−1 = 7i.
A cube root, ∛, is the number used three times: ∛64 = 4, because 4 × 4 × 4 = 64. Unlike square roots, negative numbers have real cube roots: ∛−8 = −2.
Source OpenStax · Intermediate Algebra 2e, 8.1 Simplify Expressions with Roots ↗OpenStax · Algebra and Trigonometry 2e, 2.4 Complex Numbers ↗Common Core State Standards · Grade 8, square roots and cube roots (8.EE.A.2) ↗
How to simplify a square root
Here it is with your numbers:
- √50 = √(25 × 2)
- √50 = √25 × √2
- √50 = 5√2
- √50 = 7.0711
It means 5√2 multiplied by itself is exactly 50.
A square root is simplified when no perfect square (4, 9, 16, 25 …) divides the number under it. Find the largest one, write the number as that square times what is left, and take its root out in front. For a decimal, write it over 100 (or 10,000) first.
Source OpenStax · Elementary Algebra 2e, 9.2 Simplify Square Roots ↗
Perfect squares and cubes
The square roots and cube roots that come out as whole numbers. Tap a number to try it.
| Number | Square | Cube |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1,000 |
| 11 | 121 | 1,331 |
| 12 | 144 | 1,728 |
| 13 | 169 | 2,197 |
| 14 | 196 | 2,744 |
| 15 | 225 | 3,375 |
Simplified square root examples
| Square root | Simplified | Decimal |
|---|---|---|
| √48 | 4√3 | 6.9282 |
| √50 | 5√2 | 7.0711 |
| √72 | 6√2 | 8.4853 |
| √75 | 5√3 | 8.6603 |
| √288 | 12√2 | 16.9706 |
| √500 | 10√5 | 22.3607 |
FAQs
How to calculate a square root?
Look for the number that, multiplied by itself, gives your number: √144 is 12, because 12 × 12 = 144. If there isn’t a whole one, take out the biggest perfect square factor to simplify it (√50 = √25 × √2 = 5√2), and see which whole numbers it sits between: √50 is between 7 and 8, because 49 < 50 < 64.
Source OpenStax · Elementary Algebra 2e, 9.2 Simplify Square Roots ↗
What does 2 √ 5 mean?
2√5 means 2 × √5, which is 4.4721. It is √20 simplified: √20 = √(4 × 5) = √4 × √5, and √4 is 2. We always write the whole number in front of the square root.
How do you simplify √72?
√72 = 6√2. The biggest perfect square that divides 72 is 36, so √72 = √(36 × 2) = √36 × √2 = 6√2, which is 8.4853.
How do you simplify √50?
√50 = 5√2. 25 is a perfect square that divides 50, so √50 = √(25 × 2) = √25 × √2 = 5√2, which is 7.0711.
How do I calculate cube root?
Find the number that, used three times in a multiplication, gives your number: ∛27 is 3, because 3 × 3 × 3 = 27, and ∛64 is 4. Negative numbers have cube roots too: ∛−8 is −2.
Source OpenStax · Intermediate Algebra 2e, 8.1 Simplify Expressions with Roots ↗
How is √2 calculated?
√2 is irrational: its decimals never end or repeat, so it can only be rounded: 1.4142 to 4 decimal places. 1.41 × 1.41 is 1.9881 and 1.42 × 1.42 is 2.0164, so √2 is between 1.41 and 1.42; each extra digit is found the same way.
Source Common Core State Standards · Grade 8, square roots and cube roots (8.EE.A.2) ↗
Sources
- OpenStax: Elementary Algebra 2e, 9.2 Simplify Square Roots
- OpenStax: Intermediate Algebra 2e, 8.1 Simplify Expressions with Roots
- OpenStax: Algebra and Trigonometry 2e, 2.4 Complex Numbers
- Common Core State Standards: Grade 8, square roots and cube roots (8.EE.A.2)