Radical Calculator: Master Square Roots and Nth Root Simplification
√72 doesn't have to stay √72. Let's turn those messy radicals into clean, simplified expressions you can actually work with.
You're in algebra class, staring at √50. Your teacher says "simplify it." You know √49 = 7 and √64 = 8, so √50 is somewhere between those... but that's not what "simplify" means.
Simplifying radicals isn't about calculating decimal approximations. It's about rewriting expressions in their cleanest, most usable form. And once you understand the pattern, it's actually straightforward.
Let's break down exactly how to simplify radicals—square roots, cube roots, nth roots—and why it matters beyond just getting the right answer on your homework.
What Exactly Are Radicals?
A radical is just the inverse operation of an exponent. If squaring a number means multiplying it by itself, taking the square root is asking: "What number, when squared, gives me this?"
Anatomy of a Radical:
- Radical symbol (√): Indicates we're taking a root
- Index (small ³): Which root we're taking (2 = square root, 3 = cube root, etc.)
- Radicand (27): The number under the radical
Note: When the index is 2 (square root), we usually don't write it. So √16 is understood to mean ²√16.
How to Simplify Square Roots (The Core Method)
The key to simplifying radicals is finding perfect square factors. Here's the step-by-step process:
The 3-Step Method:
- 1Find the largest perfect square that divides the radicand
Perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100...
- 2Split the radical into two parts
√(perfect square × other factor)
- 3Simplify the perfect square and keep the rest under the radical
Example: Simplify √72
Step 1: Find perfect square factors of 72
72 = 36 × 2 (36 is a perfect square!)
Step 2: Split the radical
√72 = √(36 × 2) = √36 × √2
Step 3: Simplify
√36 × √2 = 6√2
Pro Tip: Prime Factorization
Can't spot the perfect square? Break the number into prime factors and pair them up. Each pair of identical primes can come out of the radical.
√72 = √(2×2×2×3×3) = 2×3×√2 = 6√2
Practice Examples: From Simple to Tricky
√50
50 = 25 × 2
√50 = √(25 × 2) = √25 × √2 = 5√2
√48
48 = 16 × 3
√48 = √(16 × 3) = √16 × √3 = 4√3
√200
200 = 100 × 2
√200 = √(100 × 2) = √100 × √2 = 10√2
√18
18 = 9 × 2
√18 = √(9 × 2) = √9 × √2 = 3√2
√17
17 is prime—no perfect square factors
√17 stays as √17 (already simplified)
Cube Roots and Higher Roots
The same logic applies to cube roots, fourth roots, and beyond—you just look for perfect cubes (or perfect fourth powers, etc.) instead of perfect squares.
Cube Roots (∛)
Perfect cubes: 8, 27, 64, 125, 216...
Example: ∛54
54 = 27 × 2
∛54 = ∛(27 × 2) = ∛27 × ∛2 = 3∛2
Fourth Roots (⁴√)
Perfect fourth powers: 16, 81, 256, 625...
Example: ⁴√48
48 = 16 × 3
⁴√48 = ⁴√(16 × 3) = ⁴√16 × ⁴√3 = 2⁴√3
Adding, Subtracting, and Multiplying Radicals
Once you've simplified, you can combine radicals—but only if they have the same radicand.
Adding/Subtracting (Like Terms Only)
3√2 + 5√2 = 8√2 (same radicand)
7√3 - 2√3 = 5√3
3√2 + 5√3 = cannot simplify (different radicands)
Multiplying Radicals
√a × √b = √(a × b)
Example: √3 × √12 = √(3 × 12) = √36 = 6
Example: 2√5 × 3√2 = (2 × 3) × (√5 × √2) = 6√10
Where You'll Actually Use This
Radical simplification isn't just academic busywork. It shows up in surprisingly practical places:
- Pythagorean Theorem: Finding diagonal distances often produces radical answers (√2, √5, etc.). Simplified radicals are cleaner and easier to compare.
- Physics formulas: Velocity, acceleration, and energy calculations frequently involve square roots. Simplified forms make unit analysis clearer.
- Engineering: Electrical impedance, signal processing, and structural calculations use radicals extensively.
- Financial models: Standard deviation and volatility calculations produce square roots that need simplification for interpretation.
- Computer graphics: Distance formulas, normalization of vectors, and 3D math all rely on efficient radical computations.
Common Mistakes to Avoid
√(a + b) ≠ √a + √b
You can't split addition/subtraction under a radical!
Wrong: √(9 + 16) = √9 + √16 = 3 + 4 = 7
Right: √(9 + 16) = √25 = 5
Forgetting Negative Roots
When solving equations, remember ±√
If x² = 9, then x = ±3 (not just 3)
Leaving Perfect Squares Inside
If you can pull it out, you should. √32 should be 4√2, not left as √32.
The Bottom Line
Simplifying radicals is pattern recognition. Find the perfect squares (or cubes, or fourth powers), pull them out, and leave the rest inside. That's it.
Once you master the technique, you'll be able to simplify any radical expression—whether it's √200 in your algebra homework or the standard deviation formula in your statistics class.
And more importantly, you'll understand why simplified radicals are cleaner, easier to work with, and often reveal patterns that messy unsimplified forms hide.