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

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.