# Square Root of 12 - Value, Simplification, Examples

### TL;DR
The square root of 12 (\(\sqrt{12}\)) equals \(2\sqrt{3}\) in simplest radical form, or approximately 3.464 in decimal. This article covers the simplification step, the long-division decimal computation, where \(\sqrt{12}\) appears in geometry, and the common mistakes when simplifying radicals.

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The **square root of 12** simplifies to \(2\sqrt{3} \approx 3.464\). Unlike \(\sqrt{64}\) (which is the integer 8), \(\sqrt{12}\) is irrational — but its radical form simplifies because 12 has a perfect-square factor of 4.

### Quick Answer:
- **Result:** \(\sqrt{12} = 2\sqrt{3} \approx 3.464\)
- **Notation:**  \(2\sqrt{3}\) (simplest radical form), or 3.464 to three decimal places
- **Method shown:** Factoring out a perfect square
- **Exact form:**  \(2\sqrt{3}\) — irrational but simplifiable because \(12 = 4 \times 3\)
- **Irrational?** Yes — \(\sqrt{3}\) is irrational, so \(2\sqrt{3}\) is too

## Quick Reference Table

| Number \(n\) | \(n\sqrt{n}\) (simplified) | \(n\sqrt{n}\) (decimal) |
| --- | --- | --- |
| 8 | \(2\sqrt{2}\) | 2.828 |
| 10 | \(\sqrt{10}\) (already simplified) | 3.162 |
| **12** | \(2\sqrt{3}\) | **3.464** |
| 18 | \(3\sqrt{2}\) | 4.243 |
| 20 | \(5\sqrt{5}\) | 4.472 |
| 27 | \(3\sqrt{3}\) | 5.196 |
| 32 | \(4\sqrt{2}\) | 5.657 |
| 48 | \(4\sqrt{3}\) | 6.928 |
| 50 | \(5\sqrt{2}\) | 7.071 |
| 75 | \(5\sqrt{3}\) | 8.660 |

## Where \(\sqrt{12}\) Appears

\(\sqrt{12}\) shows up as the **height of an equilateral triangle with side length 4**. For an equilateral triangle of side \(s\), the height is \(h = \frac{s\sqrt{3}}{2}\) — and with \(s=4\), \(h = \frac{4\sqrt{3}}{2} = 2\sqrt{3} = \sqrt{12}\). It also appears as the **diagonal of a 2×8 rectangle** by the Pythagorean theorem, and in trigonometry as \(4 \cos(30°) = 2\sqrt{3}\).

## Why \(\sqrt{12}\) Simplifies — and \(\sqrt{10}\) Doesn't

A square root simplifies when the number under the radical has a perfect-square factor greater than 1. Compare:
- \(12=4\times3\) — has perfect-square factor 4, so \(\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}\).
- \(10=2\times5\) — no perfect-square factor greater than 1, so \(\sqrt{10}\) doesn't simplify.

The general rule:
\(a \cdot b^2 = b\sqrt{a}\) 
Pull every perfect-square factor out, leaving only the non-square part inside.

## How to Simplify \(\sqrt{12}\) — Step by Step

Step 1: Find the largest perfect-square factor of 12.
Factors of 12: 1,2,3,4,6,12. The perfect squares among these: 1,4. The largest perfect-square factor is 4.

Step 2: Rewrite 12 as \(4 \times 3\).

Step 3: Use the product property of square roots.
\(\sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}\)

Step 4: Check — can \(\sqrt{3}\) be simplified further? No perfect-square factor greater than 1. So \(2\sqrt{3}\) is simplest form.

\(\sqrt{12} = 2\sqrt{3}\)

## The Decimal Value — Long Division
If you need a decimal approximation: \(\sqrt{3} \approx 1.7320508\ldots\), so \(\sqrt{12} = 2\sqrt{3} \approx 3.4641016\ldots\)

## Is the Square Root of 12 Rational or Irrational?

\(\sqrt{12}\) is **irrational**. It cannot be written as a fraction: \(\frac{p}{q}\) of two integers, and its decimal expansion (3.4641016…) neither terminates nor repeats.

**Why.** A whole number has a rational square root only if it's a perfect square. 12 sits between 9 and 16 — not a perfect square. So \(\sqrt{12}\) is irrational.

**The simplified form makes this visible.** \(\sqrt{12} = 2\sqrt{3}\). The 2 is rational, but \(\sqrt{3}\) is irrational. A rational number times an irrational number is always irrational.

**Practical takeaway.** Leave it as \(2\sqrt{3}\) unless a decimal is specifically asked for.

## Common Mistakes With Square Root of 12

### Mistake 1: Not simplifying \(\sqrt{12}\) to \(2\sqrt{3}\)
**Where it slips in:** Final answers in geometry and algebra problems. 
**Don't do this:** Leaving \(\sqrt{12}\) as the final answer.

### Mistake 2: Pulling out a non-perfect-square factor
**Don't do this:** Incorrect simplification by factoring incorrectly.

### Mistake 3: Writing \(2\sqrt{3}\) as \(6\sqrt{6}\)
**Can't substitute the numbers inside the radicals, be careful with the differences.**

## Frequently Asked Questions

What is \(\sqrt{12}\) in simplest form?  \(\sqrt{12} = 2\sqrt{3}\)  
What is \(\sqrt{12}\) in decimal? Approximately 3.464\ldots  
Is \(\sqrt{12}\) rational or irrational? Irrational.  
How do I simplify \(\sqrt{12}\)? Find the largest perfect-square factor of 12. That's 4.  
Is \(\sqrt{12}\) the same as \(6 \cdot 2\sqrt{6}\)? Yes, but not in simplest form.  
What is \(\sqrt{12}\) in fraction form? Can't be written exactly as a fraction.