Square Roots and Cube Roots: A Clear Guide
Square roots and cube roots are building blocks of mathematics that appear in geometry, physics, engineering and statistics. They are the inverse of squaring and cubing a number. Once you understand what they are asking, they become far less intimidating.
What Is a Square Root?
The square root of a number is the value that, when multiplied by itself, gives that number.
√25 = 5 because 5 × 5 = 25. √81 = 9 because 9 × 9 = 81. √144 = 12 because 12 × 12 = 144.
Every positive number has two square roots: one positive and one negative. By convention, the radical sign refers to the positive root.
Perfect Squares
A perfect square is a whole number whose square root is also a whole number.
| Number | How formed | Square root |
|---|---|---|
| 1 | 1² | 1 |
| 4 | 2² | 2 |
| 9 | 3² | 3 |
| 16 | 4² | 4 |
| 25 | 5² | 5 |
| 49 | 7² | 7 |
| 64 | 8² | 8 |
| 100 | 10² | 10 |
| 144 | 12² | 12 |
Irrational Square Roots
Most numbers are not perfect squares. Their square roots are irrational — they continue forever without repeating.
√2 ≈ 1.41421... √3 ≈ 1.73205... √5 ≈ 2.23607...
These cannot be written as exact fractions. They are expressed as simplified radicals or rounded decimals.
How to Estimate a Square Root by Hand
Find the two perfect squares the number sits between. Example: estimate √50. 49 = 7². 64 = 8². So √50 is between 7 and 8. Since 50 is very close to 49, √50 ≈ 7.07.
What Is a Cube Root?
The cube root of a number is the value that, when multiplied by itself three times, gives that number.
∛8 = 2 because 2³ = 8. ∛27 = 3 because 3³ = 27. ∛64 = 4 because 4³ = 64.
Unlike square roots, cube roots of negative numbers exist and are negative. ∛(−8) = −2 because (−2)³ = −8.
Perfect Cubes
| Calculation | Cube root |
|---|---|
| 1³ = 1 | ∛1 = 1 |
| 2³ = 8 | ∛8 = 2 |
| 3³ = 27 | ∛27 = 3 |
| 4³ = 64 | ∛64 = 4 |
| 5³ = 125 | ∛125 = 5 |
| 10³ = 1,000 | ∛1,000 = 10 |
Simplifying Radical Expressions
A square root can often be simplified by factoring out any perfect square from under the radical sign.
Example: Simplify √72
Factor out the largest perfect square: 72 = 36 × 2
√72 = √36 × √2
Answer: 6√2
How Square and Cube Roots Appear in Real Life
- Square roots appear in the Pythagorean theorem, distance formulas, area calculations and standard deviation
- Cube roots appear in volume calculations — the side length of a cube is the cube root of its volume
- Engineers use roots when calculating magnitudes, signal amplitudes and material stress
- Finance uses square roots in risk and volatility formulas
Use the Square Root Calculator to find any root instantly. The Cube Root Calculator handles cube roots. For higher powers and nth roots, the Exponent Calculator has you covered.
Frequently Asked Questions
What is the square root of a negative number?
There is no real square root of a negative number. If you try to find √(−4), the answer does not exist in the real number system. In advanced mathematics, the square root of −1 is defined as the imaginary number i, which leads into complex numbers.
Can cube roots be negative?
Yes. The cube root of a negative number is a negative number. ∛(−27) = −3 because (−3)³ = −27. This is different from square roots, where negative inputs have no real answer.
How do I simplify a square root?
Factor out the largest perfect square from under the radical sign. For √72: find that 72 = 36 × 2, take √36 = 6 outside, leaving 6√2. The number under the radical is now as small as possible.
What is an nth root?
An nth root is the inverse of raising to the nth power. The square root is the 2nd root, the cube root is the 3rd root. The 4th root of 81 is 3, because 3⁴ = 81. Use the Exponent Calculator to compute any nth root.