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