# 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.