Square Root of 567 — Value, Simplification, and Steps
Square Root of 567 — Value, Simplification, and Steps
TL;DR
The square root of 567 is 979\sqrt{7}97 in exact form and approximately 23.812 as a decimal. This article gives the value, shows why 567\sqrt{567}567 is irrational, simplifies it by prime factorization, and works the long-division method step by step.
The square root of 567 is approximately 23.812, and in exact form it is 979\sqrt{7}97, an irrational number that never terminates or repeats.
Quick Answer:
Result: 567≈23.812\sqrt{567} \approx 23.812567≈23.812
Notation: 567=97\sqrt{567} = 9\sqrt{7}567=97 (simplest radical form)
Method shown: Prime factorization and long division
Approximate value (irrational): 23.81176180023.811761800
Exact form: 979\sqrt{7}97
Quick Reference Table
| Number | Square root (approx.) | Exact / simplified | Rational? |
|---|---|---|---|
| 529 | 23.000 | 232323 | Rational |
| 560 | 23.664 | 4354\sqrt{35}435 | Irrational |
| 567 | 23.812 | 979\sqrt{7}97 | Irrational |
| 575 | 23.979 | 5235\sqrt{23}523 | Irrational |
| 576 | 24.000 | 242424 | Rational |
| 153 | 12.369 | 3173\sqrt{17}317 | Irrational |
| 255 | 15.969 | 255\sqrt{255}255 | Irrational |
Where The Square Root of 567 Appears
567\sqrt{567}567 appears in distance problems on a coordinate grid whenever the squared gaps add to 567567567, and in any right triangle whose leg-squares sum to 567567567. Because 567=81×7567 = 81 \times 7567=81×7, it is also the diagonal of a rectangle whose sides are 999 and 969\sqrt{6}96, the kind of simplification that keeps surd arithmetic exact instead of drifting into rounding error.
What The Square Root of 567 Means
The square root of 567 is the positive number whose square is 567567567. In symbols, 567×567=567\sqrt{567} \times \sqrt{567} = 567567×567=567.
Because 232=52923^2 = 529232=529 and 242=57624^2 = 576242=576, the answer lies between 232323 and 242424, close to 242424.
Is the Square Root of 567 Rational or Irrational?
The square root of 567 is irrational. A rational number can be written as a fraction pq\frac{p}{q}qp of two integers; 567\sqrt{567}567 cannot.
The prime factorization of 567567567 is 34×73^4 \times 734×7, so the prime 777 appears to an odd power. A perfect square needs every prime in an even power, so 567567567 is not a perfect square and its root runs on forever without repeating.
How To Compute The Square Root of 567
Method 1: Prime factorization (simplest radical form)
Break 567567567 into primes:
567=3×189567 = 3 \times 189567=3×189
567=3×3×63567 = 3 \times 3 \times 63567=3×3×63
567=3×3×3×21567 = 3 \times 3 \times 3 \times 21567=3×3×3×21
567=34×7567 = 3^4 \times 7567=34×7
Every pair of 333s comes out as one 333; two pairs give 3×3=93 \times 3 = 93×3=9:
567=34×7\sqrt{567} = \sqrt{3^4 \times 7}567=34×7
567=97\sqrt{567} = 9\sqrt{7}567=97
Since 7≈2.6458\sqrt{7} \approx 2.64587≈2.6458:
9×2.6458=23.8129 \times 2.6458 = 23.8129×2.6458=23.812
Final answer: 567=97≈23.812\sqrt{567} = 9\sqrt{7} \approx 23.812567=97≈23.812
Method 2: Long division
Group the digits of 567567567 in pairs from the right: 555 and 676767.
Find the largest number whose square is at most 555: that is 222, since 22=42^2 = 422=4.
Subtract to get remainder 111, then bring down 676767 to make 167167167.
Double the quotient so far (222) to get 444; find a digit xxx so that 4x×x≤1674x \times x \le 1674x×x≤167.
Test x=3x = 3x=3: 43×3=129≤16743 \times 3 = 129 \le 16743×3=129≤167; test x=4x = 4x=4: 44×4=176>16744 \times 4 = 176 > 16744×4=176>167. So x=3x = 3x=3.
The quotient is now 232323, remainder 167−129=38167 - 129 = 38167−129=38; place a decimal point and bring down 000000 to make 380038003800.
Continue the process to reach 23.81…23.81\ldots23.81…
Final answer: 567≈23.812\sqrt{567} \approx 23.812567≈23.812
Common Mistakes With the Square Root of 567
Mistake 1: Pulling out only one pair of threes
Where it slips in: Factoring 567=9×63567 = 9 \times 63567=9×63 and taking just the 999.
Don't do this: Write 567=363\sqrt{567} = 3\sqrt{63}567=363 and stop.
The correct way: 63=9×763 = 9 \times 763=9×7 still holds a perfect square. The full factorization is 34×73^4 \times 734×7, giving 979\sqrt{7}97. The first instinct is to stop at the first square factor you spot, but 63\sqrt{63}63 still simplifies — keep going until the number under the radical has no square factor.
Mistake 2: Reading 567 as close to 576 and writing 24
Where it slips in: Noticing 567567567 is near 576=242576 = 24^2576=242.
Don't do this: Write 567=24\sqrt{567} = 24567=24.
The correct way: 242=576≠56724^2 = 576 \ne 567242=576=567, so 567<24\sqrt{567} < 24567<24; the value is 23.81223.81223.812.
Mistake 3: Miscounting powers of 3
Where it slips in: Deciding how many 333s come out of the radical.
Don't do this: Treat 343^434 as giving a single 333 outside.
The correct way: 34=(32)23^4 = (3^2)^234=(32)2, so 34=32=9\sqrt{3^4} = 3^2 = 934=32=9. Each pair of equal factors leaves one copy outside.
A Quick Way to Check Yourself
Estimate first: 567567567 sits between 529529529 and 576576576, so the root is between 232323 and 242424, and closer to 242424. Then confirm the surd form by squaring it back: (97)2=81×7=567(9\sqrt{7})^2 = 81 \times 7 = 567(97)2=81×7=567.