Square Root of 340 — Value, Simplification, Steps

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: $\sqrt{340} = 2\sqrt{85} \approx 18.439$

Notation: Simplified radical $2\sqrt{85}$; 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: $2\sqrt{85}$

Quick Reference Table of Nearby Square Roots

The table sets √340 next to nearby roots so the size and the simplification both read at a glance.

√340 and √1360 both keep the same √85 core, while √255 nearby stays fully stuck because 255 = 3 × 5 × 17 has 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 $\sqrt{4^2 + 18^2} = \sqrt{16 + 324} = \sqrt{340} = 2\sqrt{85}$. Any diagonal or distance that reduces to 340 under the root carries this value.

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 $\sqrt{4^2 + 18^2} = \sqrt{16 + 324} = \sqrt{340} = 2\sqrt{85}$. 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 $n$: $\sqrt{n} = x$ means $x^2 = n$. 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.

$340 = 2 \times 170$

$340 = 2 \times 2 \times 85$

$340 = 2^2 \times 5 \times 17$

The pair $2 \times 2$ leaves the radical as a single 2. The 5 and 17 have no partners, so they stay inside as $5 \times 17 = 85$.

$\sqrt{340} = \sqrt{2^2 \times 85}$

$\sqrt{340} = 2\sqrt{85}$

Final answer: $\sqrt{340} = 2\sqrt{85}$.

Method 2: Estimation by bracketing

Trap √340 between two perfect squares.

$18^2 = 324$

$19^2 = 361$

So $18 < \sqrt{340} < 19$. Since 340 is closer to 324, the answer is a little above 18.4.

$18.4^2 = 338.56$

$18.5^2 = 342.25$

Final answer: $\sqrt{340} \approx 18.439$, matching $2\sqrt{85} = 2 \times 9.2195 = 18.4391$.

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 $\sqrt{340} = 2\sqrt{85} = 10\sqrt{17}$ by "taking out" a 5.

The correct way: 85 = 5 × 17 has no perfect-square factor, so nothing more comes out. Students who just learned to simplify radicals often keep pulling factors that were never squared. The final form is $2\sqrt{85}$.

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 $\sqrt{340} = 4\sqrt{85}$.

The correct way: the perfect square 4 leaves the radical as $\sqrt{4} = 2$, not as 4. So $\sqrt{340} = 2\sqrt{85}$.

Mistake 3: Adding roots across a sum

Where it slips in: using √340 inside a distance-formula step.

Don't do this: claim $\sqrt{16 + 324} = \sqrt{16} + \sqrt{324} = 4 + 18 = 22$.

The correct way: the root of a sum is not the sum of the roots. Add under the root first: $\sqrt{16 + 324} = \sqrt{340} = 2\sqrt{85} \approx 18.439$.

Conclusion

• The square root of 340 is irrational, equal to $2\sqrt{85} \approx 18.439$.

• 340 factors as $2^2 \times 5 \times 17$, 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.

To take radical simplification further with a teacher, explore Bhanzu's algebra tutor or browse math classes online.

Read More

• Square root of 245 — a sibling radical that simplifies to 7√5

• Square root of 85 — the √85 core inside √340

• Squares and square roots — the rules behind pulling out square factors

• Square root of 100 — a clean perfect-square contrast

• What is a square root — the notation and definition in full

• Cube root of 100 — the same question one index higher