# Square Root of 340 — Value, Simplification, Steps  
TL;DR  
The square root of 340 simplifies to 2√85 ≈ 18.439, because 340 factors as 2² × 5 × 17, and only the perfect square 4 pulls out. This article gives the value, the full simplification, two hand methods, and the mistakes students hit with √340.  
  
The square root of 340 is approximately **18.439**, and in exact form it is **2√85**. Only one perfect square hides inside 340 — a factor of 4 — which leaves the radical as 2 with an 85 still trapped under the root.  
  
> **Quick Answer:**  
> **Result:** 340=2√85≈18.439  
> **Notation:** Simplified radical 2√85; decimal 18.4391 (to 4 dp)  
> **Method shown:** Prime factorisation to extract the perfect-square factor, then estimation  
> **Approximate value:** 18.4391 (irrational, non-terminating)  
> **Exact form:** 2√85  
  
## Quick Reference Table of Nearby Square Roots  
| Number | Square root | Simplified form |  
| --- | --- | --- |  
| 324 | 18  | Exact (perfect square) |  
| 340 | 18.439 | 2√85 |  
| 361 | 19 | Exact (perfect square) |  
| 85 | 9.220 | √85 |  
| 1360 | 36.878 | 2√85 |  
| 255 | 15.969 | √255 (no square factor) |  
  
## Where the Square Root of 340 Appears  
A square root gives the side of a square from its area, so √340 is the side of a square holding 340 square units. It also appears through the distance formula: the distance between the points (0,0) and (4,18) is √(42 + 182) = √(16 + 324) = √(340) = 2√85. Any diagonal or distance that reduces to 340 under the root carries this value.  
  
## What a Square Root Means  
The square root of a number n is the value that multiplied by itself gives n: n=x means x²=n. When n is a perfect square, the root is a whole number; otherwise the root is **irrational**, with a decimal that never ends and never repeats.  
340 is not a perfect square, so √340 is irrational. It still simplifies partway, though, because 340 contains the perfect square 4.  
  
## How to Compute the Square Root of 340  
### **Method 1: Prime factorisation (the simplification)**  
Factor 340 into primes and look for pairs.  
340 = 2 × 170  
340 = 2 × 2 × 85  
340 = 2² × 5 × 17  
The pair 2×2 leaves the radical as a single 2. The 5 and 17 have no partners, so they stay inside as 5 × 17 = 85.  
340 = 2² × 85  
√340 = √(2² × 85)  
340 = 2√85.  
**Final answer:** 340 = 2√85.  
  
### **Method 2: Estimation by bracketing**  
Trap √340 between two perfect squares.  
18² = 324  
19² = 361  
So 18 < √340 < 19. Since 340 is closer to 324, the answer is a little above 18.4.  
**Final answer:** 340 ≈ 18.439.  
  
## Common Mistakes With Square Root of 340  
### **Mistake 1: Over-simplifying the leftover**  
**Where it slips in:** trying to break 85 down further after pulling out the 4.  
**Don't do this:** write 340=2√85=10√17 by "taking out" a 5.  
**The correct way:** 85 = 5 × 17 has no perfect-square factor, so nothing more comes out. The final form is 2√85.  
  
### **Mistake 2: Extracting the factor instead of its root**  
**Where it slips in:** knowing 4 comes out but writing the 4 itself.  
**Don't do this:** write 340=4√85.  
**The correct way:** the perfect square 4 leaves the radical as 4=2√4=2, not as 4. So 340=2√85.  
  
### **Mistake 3: Adding roots across a sum**  
**Where it slips in:** using √340 inside a distance-formula step.  
**Don't do this:** claim 16+324=√16 + √324.  
**The correct way:** the root of a sum is not the sum of the roots. Add under the root first: 16+324=√340.  
  
## Conclusion  
- The **square root of 340** is irrational, equal to 2√85 ≈ 18.439.  
- 340 factors as 2² × 5 × 17, so only the perfect square 4 pulls out, leaving √85 inside.  
- The value sits between 18 and 19 because 340 lies between the squares 324 and 361.  
- The leftover √85 cannot be reduced, since 85 has no perfect-square factor.