# Square Root of 14 — Value, How to Find It, and Examples

TL;DR

The square root of 14 is about 3.7416574, and because 14 = 2 × 7 has no repeated prime factor, \( \sqrt{14} \) is irrational and already in simplest radical form. This article gives the exact and decimal value, two by-hand methods, where \( \sqrt{14} \) shows up, and the slips students make most.

## The Answer At A Glance

> **Result:** \( \sqrt{14} \approx 3.7416574 
> **Notation:** Decimal approximation; exact form is \( \sqrt{14} \).
> **Method shown:** Long division by hand, cross-checked with estimation between perfect squares.
> **Approximate value:** 3.7417 (4 d.p.)
> **Exact form:** \( \sqrt{14} \) — cannot be simplified, since 14 = 2 × 7 has no square factor.

## Quick Reference Table — Square Roots From 5 to 24

| n | \( n \sqrt{n} \) (exact) | \( n \sqrt{n} \) (4 d.p.) |
|---| --- | --- |
| 5 | \( 5 \sqrt{5} \) | 2.2361 |
| 9 | 3 | 3.0000 |
| 10 | \( 10 \sqrt{10} \) | 3.1623 |
| 11 | \( 11 \sqrt{11} \) | 3.3166 |
| 12 | \( 2\sqrt{3} \) | 3.4641 |
| 13 | \( 13 \sqrt{13} \) | 3.6056 |
| **14** | \( 14 \sqrt{14} \) | 3.7417 |
| 15 | \( 15 \sqrt{15} \) | 3.8730 |
| 16 | 4 | 4.0000 |
| 18 | \( 3\sqrt{2} \) | 4.2426 |
| 20 | \( 2\sqrt{5} \) | 4.4721 |
| 24 | \( 2\sqrt{6} \) | 4.8990 |

\( \sqrt{14} \) sits between \( 3 \) and \( 4 \), a little past the midpoint because 14 is closer to 16 than to 9.

## Where √14 Appears

\( \sqrt{14} \) is the hypotenuse of a right triangle with legs \( 5 \) and \( 3 \), since \( (5)^2 + (3)^2 = 14 \). It is also the space diagonal of a box measuring \( 1 \times 2 \times 3 \) — Pythagoras in three dimensions gives \( \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \).

## What "square root of 14" Means

The square root of a non-negative number \( n \) is the value \( x \) such that \( x^2 = n \). For \( \sqrt{14} \), it is the positive \( x \) with \( x^2 = 14 \).

### Is The Square Root of 14 Rational or Irrational?

\( \sqrt{14} \) is **irrational**. Its prime factorization is \( 14 = 2 × 7 \) — no prime sits at an even power and \( \sqrt{14} \) is not a perfect square. Thus, it cannot be simplified into a smaller radical.

## How To Find √14 — Two Methods

### Method 1 — Long division (digit by digit)

Write 14 as 14.000000 and pair the digits.

Step 1. The largest integer whose square is at most 14 is 3 (\( 3^2 = 9 \)). Subtract: 14 − 9 = 5. Bring down 000000 to get 500.

Step 2. Double the quotient 3 to get 6. Find \( d \) with \( (60+d)⋅d≤500 \). Here, \( d=7 \) gives 67⋅7=469. Subtract: 500 − 469 = 31. Bring down 000000 to get 3100.

Continuing produces 3.7416…

**Final answer:** \( \sqrt{14} \approx 3.7417.

### Method 2 — Estimation between perfect squares

Since \( 3^2 = 9 \) and \( 4^2 = 16 \), start at 3.7: \( 3.7^2 = 13.69 \). Try 3.74: \( 3.74^2 = 13.9876 \). Each refinement closes in on 14, settling at 3.742.

## Common Mistakes With √14

### Mistake 1: Trying to simplify a non-square radicand

**Where it slips in:** A student factorizes \( 14 = 2 × 7 \) and tries to take one factor out of the root.

**Don't do this:** It stays as \( \sqrt{14} \).

### Mistake 2: Splitting the root over addition

**Where it slips in:** \( \sqrt{5 + 9} \).

**The correct way:** \( \sqrt{5 + 9} = \sqrt{14} \).

### Mistake 3: Confusing \( -\sqrt{14} \) with \( \sqrt{-14} \)

**The correct way:**  \( -\sqrt{14} \) is real; \( \sqrt{-14} \) is imaginary.

## Examples of Square Root of 14

### Example 1

**Confirm that \( \sqrt{14} \) does not simplify.**

### Example 2

**Find the space diagonal of a 1×2×3 box.**

**Correct approach:** \( \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \approx 3.742.

### Example 3

**Evaluate \( 14×\sqrt{14} \times \sqrt{14} \).**

### Example 4

**Rationalize \( \frac{7}{\sqrt{14}} \).**

### Example 5

**A square has area 14 square centimeters. Find its side length.**

Side = \( \sqrt{14} \approx 3.74 \) cm.

## Conclusion

- The **square root of 14** is approximately 3.742 — irrational, non-terminating, non-repeating.
- 14 = 2 × 7 has no repeated prime factor, so \( \sqrt{14} \) cannot be simplified.