Cube Roots Explained- How to Write and Calculate Them
What Is a Cube Root?
A cube root is the number that, when multiplied by itself three times, gives you the original number. If āx = y, then y Ć y Ć y = x.
Simple example: the cube root of 27 is 3 because 3 Ć 3 Ć 3 = 27.
That's it. Nothing fancy. Just working backwards from a cube.
How to Write Cube Roots
You use the radical symbol with a small 3 written above it. It looks like this: ā
The notation for the cube root of 64 is:
ā64 = 4
You can also write it using fractional exponents:
641/3 means exactly the same thing as ā64
The difference between a square root and a cube root matters:
- Square root (ā) ā asks "what number times itself twice equals x?"
- Cube root (ā) ā asks "what number times itself three times equals x?"
How to Calculate Cube Roots
You have three practical methods. Pick the one that fits your situation.
Method 1: Memorize Perfect Cubes
The fastest way for small numbers. Learn these common cubes:
| Number | Cube (nĀ³) | Cube Root ā |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 8 | 2 |
| 3 | 27 | 3 |
| 4 | 64 | 4 |
| 5 | 125 | 5 |
| 6 | 216 | 6 |
| 7 | 343 | 7 |
| 8 | 512 | 8 |
| 9 | 729 | 9 |
| 10 | 1000 | 10 |
If you know your multiplication tables, you already know most of these. š¢
Method 2: Prime Factorization
Works when the number isn't a perfect cube but can be broken down.
Example: Find ā1728
- Factor 1728 into primes: 1728 = 2Ā³ Ć 2Ā³ Ć 3Ā³
- Group the factors into triples: (2Ā³) Ć (2Ā³) Ć (3Ā³)
- Take one number from each triple: 2 Ć 2 Ć 3 = 12
- Answer: ā1728 = 12
Check: 12 Ć 12 Ć 12 = 1728 ā
Method 3: Estimation for Non-Perfect Cubes
When you encounter something like ā50, you need to estimate.
Step 1: Find the two perfect cubes it falls between
- 3Ā³ = 27 (too low)
- 4Ā³ = 64 (too high)
- So ā50 is between 3 and 4
Step 2: Get closer
Try 3.7: 3.7Ā³ = 50.653 (close)
Try 3.7: 3.7Ā³ = 50.653 (close)
Try 3.68: 3.68Ā³ = 49.87 (even closer)
Answer is approximately 3.68
For most practical purposes, a calculator handles this instantly. But knowing the process helps you understand what's actually happening.
Cube Roots of Negative Numbers
Here's something square roots can't do: cube roots work fine with negative numbers.
The cube root of -27 is -3
Why? Because (-3) Ć (-3) Ć (-3) = -27
Three negatives multiplied together always give a negative. So every real number has exactly one real cube root.
Common Mistakes to Avoid
- Confusing the exponent: Square root asks for two multiplications, cube root asks for three. Don't mix them up.
- Forgetting negative cubes: -8 has a real cube root (-2). You can't find a real square root of -8.
- Rounding too early: If you need precision, keep more decimal places during calculation.
- Overcomplicating simple problems: For perfect cubes, just ask "what number cubed gives me this?"
Getting Started: Your Quick Reference
Bookmark this. You'll need it.
To find a cube root:
- Is it a perfect cube from 1-10? ā Memorized answer
- Is it a larger number? ā Try prime factorization
- Is it messy (like ā7)? ā Estimate between nearest integers, then refine
- Is it negative? ā Find the positive cube root, then add minus sign
That's the whole game. Practice with a few problems and you'll have the basic ones memorized within a day.