Square Root of 8 — Value, Simplified Form, and Examples
Square Root of 8 — Value, Simplified Form, and Examples
TL;DR
The square root of 8 is 2√2, approximately 2.828, and it is irrational. Unlike a prime radicand, 8 simplifies because 8=4×2. This article gives the exact and decimal value, two by-hand methods, where 2√2 shows up, and the mistakes students make most.
The Answer At A Glance
Quick Answer:
Result: √8 = 2√2 ≈ 2.8284271
Notation: Exact (simplest radical) form 2√2; decimal approximation 2.8284.
Method shown: Prime factorization to simplify, then long division to confirm the decimal.
Approximate value: 2.8284 (4 d.p.)
Exact form: 2√2 — 8=2^3, so one pair of 2's comes out of the root.
Quick Reference Table — Square Roots From 1 to 20
| n | √n (exact) | √n (4 d.p.) |
|---|---|---|
| 1 | 1 | 1.0000 |
| 2 | 2√2 | 1.4142 |
| 3 | 3√3 | 1.7321 |
| 4 | 2 | 2.0000 |
| 5 | 5√5 | 2.2361 |
| 6 | 6√6 | 2.4495 |
| 7 | 7√7 | 2.6458 |
| 8 | 2√2 | 2.8284 |
| 9 | 3 | 3.0000 |
| 10 | 10√10 | 3.1623 |
| 11 | 11√11 | 3.3166 |
| 12 | 12 | 3.4641 |
| 13 | 13√2 | 3.6056 |
| 14 | 14√7 | 3.7420 |
| 15 | 15√3 | 3.8720 |
| 16 | 4 | 4.0000 |
| 17 | 17√17 | 4.1231 |
| 18 | 18√2 | 4.2426 |
| 19 | 19√19 | 4.3589 |
| 20 | 20√5 | 4.4721 |
Where √8 Appears
√8, written 2√2, is the diagonal of a square whose side length is 2 — Pythagoras gives 2+2=√8. It is exactly twice the diagonal of a unit square (2√2), which is why 2√2 turns up whenever a 45° direction is scaled. The same value appears as the distance between the points (0,0) and (2,2) on a coordinate grid.
What 'square root of 8' Means
The square root of a non-negative number n is the value x such that x² = n. For √8, it is the positive x with x² = 8.
Because 2²=4 and 3²=9, the answer lands between 2 and 3 — and (2√2)²=4⋅2=8 confirms the simplified form is exact.
Is The Square Root of 8 Rational or Irrational?
√8 is irrational. Its prime factorization is 8=2³ — the prime 2 appears three times, an odd power, so √8 is not a perfect square.
Simplifying to 2√2 does not change that: 2√2 is itself irrational, and an integer times an irrational number stays irrational. The decimal 2.8284271… never terminates and never repeats.
How To Find √8 — Two Methods
Method 1 — Prime factorization (the simplification)
Break 8 into primes: 8=2×2×2=2²×2.
A pair of equal factors under a root comes out as a single factor: 2×2=2√2.
Then 2=2×1.41421=2.82842.
Final answer: √8 = 2√2 ≈ 2.8284.
Method 2 — Long division (digit by digit)
Write 8 as 8.000000 and pair the digits after the decimal point.
Step 1. The largest integer whose square is at most 8 is 2 (2² = 4). Subtract: 8−4=4. Bring down 000000 to get 400.
Step 2. Double the quotient 2 to get 4. Find d with (40+d)⋅d≤400. Here d=8 gives 48⋅8=384. Subtract: 400−384=16. Bring down 000000 to get 1600.
Step 3. Double 2.8 to get 5.6. Find d with (560+d)⋅d≤1600. Here d=2 gives 562⋅2=1124. Subtract: 1600−1124=476.
Continuing produces 2.8284…, matching 2√2.
Final answer: √8 ≈ 2.8284.
What are the most common mistakes with √8?
Mistake 1: Leaving √8 unsimplified
Where it slips in: A student computes the decimal but stops before simplifying the radical, so an exam answer reads "√8" where "2√2" was wanted.
Don't do this: Treating √8 as fully simplified just because it's a single root symbol.
The correct way: Check for a square factor first — 8=4×2, so √8 = 2√2.
Mistake 2: Pulling out the wrong factor
Where it slips in: Rushing the prime factorization and taking out the whole 4 instead of the square root of 4.
Don't do this: 8=4×2.
The correct way: 4×2=4, 2=2√2. Only the root of the square factor leaves the radical.
Mistake 3: Splitting the root over addition
Where it slips in: When √8 appears as 4+4 inside the diagonal calculation.
Don't do this: 4+4= \sqrt{4} + \sqrt{4}.
The correct way: 4+4=8=√8=2√2.
Examples of Square Root of 8
Example 1
Simplify √8 to radical form.
8=4⋅2=2√2 ≈ 2.828.
Example 2 (Wrong path first)
Find the diagonal of a square with side 2.
Wrong attempt. A student writes the diagonal as 4.
Correct. 2+2=8=√2² + 2² = √8 = 2√2.
Example 3
Add 8+2√8 + √2.
8=2√2, so 8+2=3√2 ≈ 4.243.
Example 4
Rationalise \frac{4}{√8}.
4=2≈1.414.
Example 5
A square garden has area 8 square metres. Find its side length.
Side =√8 = 2√2 ≈ 2.83 m.
Conclusion
- The square root of 8 is 2√2, approximately 2.828 — irrational but simplifiable.
- 8=2³, so the square factor 4 leaves the root as 2.
- Prime factorization simplifies √8; long division confirms the decimal.
- Only the root of a square factor comes out — 4×2=2√2, not 4√2.
Frequently Asked Questions
What is the value of the square root of 8?
√8 = 2√2 ≈ 2.8284271…
What is the square root of 8 in simplest radical form?
2√2, because 8=4×2.
Is the square root of 8 rational or irrational?
Irrational. 8=2³ is not a perfect square.
Is 8 a perfect square?
No. The nearest perfect squares are 4 and 9.
How is √8 different from √2?
√8 is exactly twice √2.