# Square Root of 10 - Value, Calculation, Examples

## TL;DR
The **square root of 10** (10\sqrt{10}10​) is approximately 3.16227766. This article gives the exact form, the decimal approximation to several places, the long-division method to compute it by hand, where 10\sqrt{10}10​ shows up in geometry (the diagonal of a 1×3 rectangle), and why it's irrational.

The **square root of 10** is approximately 3.16227766, an irrational number whose decimal expansion never terminates or repeats.

### Quick Answer:
- **Result:** 10≈3.16227766\sqrt{10} \approx 3.16227766
- **Notation:** 10\sqrt{10}10​ (exact, irrational), or 3.162 to three decimal places
- **Method shown:** Long division
- **Exact form:** 10\sqrt{10}10​ — cannot be simplified further (10 has no perfect-square factors greater than 1)
- **Irrational?** Yes — non-terminating, non-repeating decimal

## Quick Reference Table
| Number nnn | n\sqrt{n}n​ (approx.) | Rational or Irrational |
| --- | --- | --- |
| 1 | 1 | Rational |
| 4 | 2 | Rational |
| 9 | 3 | Rational |
| **10** | **3.162** | **Irrational** |
| 12 | 3.464 | Irrational |
| 16 | 4 | Rational |
| 25 | 5 | Rational |
| 49 | 7 | Rational |
| 64 | 8 | Rational |
| 100 | 10 | Rational |

## Where 10\sqrt{10}10​ Appears
10\sqrt{10}10​ shows up as the **diagonal of a 1×3 rectangle** — the Pythagorean theorem gives 12 + 32 = 10\sqrt{1^2 + 3^2} = \sqrt{10}. It also appears in engineering as the length of a unit step in **base-10 logarithmic scales** — a power ratio of 10 dB corresponds to a voltage ratio of 10\sqrt{10}10​.

## What Is a Square Root?
The **square root** of a number nnn is a value rrr such that r2 = n. The square root of 10 is the number that, multiplied by itself, gives 10.

There is no integer that does this. So 10\sqrt{10}10​ lies between 3 and 4, somewhere closer to 3.

Because 10 is not a perfect square, 10\sqrt{10}10​ is an **irrational number** — its decimal goes on forever without ending or repeating. The first 12 digits are: 10 = 3.16227766017…\sqrt{10} = 3.16227766017…

## Is the Square Root of 10 Rational or Irrational?
10\sqrt{10}10​ is **irrational** — meaning it cannot be written as a fraction pq of two integers, and its decimal expansion neither terminates nor repeats.

## How to Find 10\sqrt{10}10​ — Long Division Method
**Step 1:** Pair the digits from the decimal point. 10.00‾00‾00‾…

**Step 2:** Find the greatest number whose square is ≤10. That's 3. So, the first digit of the quotient will be 3.

**Step 3:** Subtract: 10 − 9 = 1. Bring down the next pair of zeros: 100.

**Step 4:** Double the current quotient: 3×2=6. Find a digit d such that (60+d)×d≤100. Try d=1: 61×1=61. Therefore, the quotient becomes 3.1.

**Step 5:** Subtract 61 from 100: remainder 39. Bring down next pair: 3900.

**Step 6:** Find d such that (620+d)×d≤3900. Try d=6: 626×6=3756. So, the quotient becomes 3.16.

**Step 7:** Continue this process to find further decimal places, ultimately yielding a longer decimal approximation.

## Common Mistakes With Square-Root-of-10
### **Mistake 1: Reporting 10 = ±3.162**
Where it slips in: Solving x2 = 10.

### **Mistake 2: Trying to simplify 10\sqrt{10}10​**
The correct way: 10\sqrt{10}10​ is already in simplest radical form.

### **Mistake 3: Rounding too early**
Keep 10\sqrt{10}10​ as 10\sqrt{10}10​ symbolically until the final step.

## Frequently Asked Questions
### What is the value of 10\sqrt{10}10​?
10≈3.16227766.

### Is 10\sqrt{10}10​ rational or irrational?
Irrational.

### What is the square of 10\sqrt{10}10​?
(10)2=10(\sqrt{10})^2 = 10.

### Can 10\sqrt{10}10​ be simplified?
No, it is already in simplest form.

### Where does 10\sqrt{10}10​ appear in geometry?
As the diagonal of a 1×3 rectangle.