Square Root of 72 in Radical Form — What It Actually Is and Why It Matters
Quick question: if someone hands you √72 and asks you to simplify it, do you freeze? That's why most people do. And it's not because it's hard — it's because nobody ever explained why you do what you do. They just said "factor out the square" and moved on.
Here's the thing — the square root of 72 in radical form isn't just a math exercise. But it's a pattern. Also, once you see it, you'll never stumble on √72 again. And honestly, the same logic works for dozens of other radicals. Let's walk through it like real people.
What Is the Square Root of 72?
The square root of 72 is the number that, when multiplied by itself, gives you 72. On a calculator, that's about 8.485. But that's a decimal approximation. In math class, you're usually not asked for a decimal. You're asked to express it in radical form* — meaning simplify the square root so what's under the radical symbol (the "radicand") is as small as possible.
So the goal isn't to get a number. The goal is to rewrite √72 as something cleaner, like 6√2.
But how do you get there? You need to understand what "simplifying a radical" actually means.
What "Radical Form" Really Means
Radical form just means the answer is written using the radical symbol (√). Here's the thing — if the radicand — the number under the symbol — has any perfect square factors, you can pull them out. That's the whole game.
So √72 in radical form, fully simplified, is 6√2.
Why? Because 72 = 36 × 2, and 36 is a perfect square. The square root of 36 is 6. That 6 comes out, and you're left with √2 underneath.
That's it. That's the trick.
Why Simplifying Radicals Matters
You might be thinking, "Okay cool, so I can write √72 as 6√2. So what?" Fair question.
It's the standard. In nearly every math class past middle school, if you leave √72 in unsimplified form on a test, you'll lose points. Teachers want the simplest radical form. Period.
It makes other math easier. When you're adding, subtracting, or comparing radicals, you can't combine √72 with √2 unless you simplify first. They're considered "unlike" radicals until they match.
It shows up everywhere. Geometry (finding side lengths of right triangles), physics (velocities and accelerations), engineering, computer graphics — simplified radicals are the default working form. You don't want √72 cluttering up your equation when 6√2 does the job in fewer characters.
So yeah. Practically speaking, it's not busywork. It's the way professionals write math.
How to Simplify √72 Step by Step
Here's the process that works for any radical, not just √72. Once you learn it, you can simplify √75, √180, √200 — all of them.
Step 1: Find the Prime Factorization
Break 72 down into prime numbers:
72 = 2 × 2 × 2 × 3 × 3
Or written with exponents: 72 = 2³ × 3²
We're talking about the foundation of everything else. Without it, you're guessing.
Step 2: Look for Pairs
Here's the key rule: for every pair of identical factors, one comes out of the radical. A pair of the same number is a perfect square.
In 2³ × 3², you've got:
- Three 2s. That gives you one pair of 2s (with one 2 left over).
- Two 3s. That's one pair of 3s.
Step 3: Pull the Pairs Out
Each pair becomes a single number outside the radical:
- One pair of 2s → 2 (one 2 comes out)
- One pair of 3s → 3 (one 3 comes out)
Multiply them: 2 × 3 = 6
Step 4: Leave the Leftovers Under the Radical
After pulling out the pairs, what's left under the radical?
- One 2 (the unpaired leftover from the 2³)
So you get: 6√2
And that's the square root of 72 in simplest radical form.
A Faster Shortcut (Once You Get Comfortable)
Let me be honest — the prime factorization is the teach* method. It's how you learn what's actually happening. But once you've done it a few dozen times, you'll start spotting perfect square factors at a glance.
Continue exploring with our guides on what do smelling salts feel like and penicillin was discovered and isolated from a.
For √72, ask yourself: what's the biggest perfect square that divides into 72?
- 4? Yes. 72 ÷ 4 = 18. So √72 = 2√18. Better, but not done.
- 9? Yes. 72 ÷ 9 = 8. So √72 = 3√8. Still not done.
- 36? Yes! 72 ÷ 36 = 2. So √72 = √36 × √2 = 6√2. Done.
The 36 route is the shortest. Once you can spot it, you'll skip the prime factorization entirely.
In practice, this is what most math teachers actually want you to do once you've shown you understand the underlying logic. So learn the long way, then get fast.
Common Mistakes People Make With √72
I've graded enough homework (and made enough of my own mistakes) to know where people get tripped up. Here are the big ones.
Forgetting to Simplify All the Way
A lot of students will write √72 = 6√2 and stop. But always check: is the radicand as small as it can be? They're not. That's correct! √18 can still be simplified (to 3√2). But others will write √72 = 2√18 and think they're done. If it has any perfect square factors left, keep going.
Pulling Out Numbers That Aren't Pairs
This one shows up constantly. You'll see someone write √72 = 4√… something, because 4 is a square. But you can't just invent* a 4. On the flip side, the numbers outside the radical have to come from actual pairs inside the radicand. * Otherwise you're breaking math.
Mixing Up "Radical Form" and "Decimal Form"
The problem says radical form*. That means the answer should have a √ in it. If you write 8.Think about it: 485, you've technically found the value, but you haven't answered the question. Always read carefully.
Forgetting to Multiply Outside When You Pull a Pair
Say you spotted that 72 = 4 × 18, and pulled the √4 out as 2. But then you have 2√18, and you keep simplifying — √18 = 3√2. Now you need to multiply that 3 with the 2 you already pulled out. So 2 × 3 = 6, and you get 6√2. On the flip side, good. People forget this multiplication step and leave an answer like 2·3√2, which isn't wrong but isn't simplified either.
Practical Tips for Simplifying Any Radical
Here's what actually works in real problem-solving situations, whether you're doing homework, studying for a test, or just trying to remember how this stuff goes.
Memorize the first dozen perfect squares. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169. The more you recognize, the faster you go.
Always look for the biggest perfect square first. It saves steps. If 72 had a factor of 144, you'd be done in one move. So scan for the largest square, not the smallest.
When in doubt, use prime factorization. It's slower, but it's foolproof. If you can't eyeball it, factor it out.
Double-check by squaring your answer. 6√2 squared is 36 × 2 = 72. If your answer squares back to 72, you've done it right.
Write neatly. Seriously. A missed pair of 2s in your prime factorization is a hard mistake to catch if your work is a mess. Clean work = fewer errors.
FAQ
What is the square root of 72 in simplest radical form?
6√2. You get this by factoring 72 into 36 × 2, taking the square root of 36 (which is
which is 6), and leaving √2 as is.
How do you know you've simplified it enough?
You've simplified it enough when the number inside the square root (the radicand) has no more perfect square factors. Practically speaking, in 6√2, the number under the radical is 2. The only factors of 2 are 1 and 2, and neither is a perfect square (other than 1, which is trivial and doesn't help). So, it's fully simplified.
Why is simplifying radicals useful?
It's useful for a few reasons. To give you an idea, it's much clearer that 6√2 is half of 12√2 than it is that √72 is half of √288. It makes answers easier to read and compare. It's also essential for more advanced math, like algebra and geometry, where simplified radical forms are needed to combine like terms or solve equations correctly.
Mastering the simplification of radicals like √72 isn't just about getting one problem right; it's about building a solid foundation for all future math. The next time you see a messy radical under a square root sign, remember the steps: factor, find the perfect square, pull it out, and multiply. That's why it's a skill that trains you to look for patterns, be meticulous, and check your work—habits that pay off far beyond the classroom. With a little practice, it becomes second nature, and you'll see the math become clearer, one simplified radical at a time.
