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# Square Root of 255 — Value, Simplification, and Steps

## TL;DR
The square root of 255 is approximately **15.969** and cannot be simplified, because 255 has no perfect-square factor, so \( \sqrt{255} \) is already in simplest form. This article gives the value, shows why it is irrational, and works the long-division method step by step.

## Quick Reference Table

| Number | Square root (approx.) | Exact / simplified       | Rational? |
| ---    | ---                   | ---                      | ---       |
| 225    | 15.000                | 15                       | Rational  |
| 245    | 15.652                | \( 75\sqrt{5} \)       | Irrational|
| 253    | 15.906                | \( 253 \sqrt{253} \)   | Irrational|
| 255    | 15.969                | \( \sqrt{255} \)       | Irrational|
| 256    | 16.000                | 16                       | Rational  |
| 153    | 12.369                | \( 317\sqrt{17} \)     | Irrational|
| 567    | 23.812                | \( 97\sqrt{7} \)       | Irrational|

## Where The Square Root of 255 Appears
\( \sqrt{255} \) turns up in distance calculations on a coordinate grid whenever the squared horizontal and vertical gaps add to 255. It also sits one step below 256, which is a number every programmer meets because it is \( 2^8 \), the count of values a single byte can hold, so \( \sqrt{255} \) is just under the diagonal reach of a 16x16 square.

## What the Square Root of 255 Means
The **square root of 255** is the positive number whose square is 255. In symbols, \( \sqrt{255} \times \sqrt{255} = 255 \).

Because \( 15^2 = 225 \) and \( 16^2 = 256 \), the answer lands between 15 and 16, very close to 16.

## Is the Square Root of 255 Rational or Irrational?
The square root of 255 is **irrational**. A rational number can be written as a fraction \( \frac{p}{q} \) of two integers; \( \sqrt{255} \) cannot.

The prime factorization of 255 is \( 3 \times 5 \times 17 \): three different primes, each appearing once. No prime appears in a pair, so there is no perfect-square factor and 255 is not a perfect square. Its decimal runs on forever without repeating.

## How to Compute the Square Root of 255

### Method 1: Prime factorization (check for simplification)
Break 255 into primes:
\[
255 = 3 \times 85
\]
\[
255 = 3 \times 5 \times 17
\]
Every prime appears once, so nothing can leave the radical:
\[
\sqrt{255} = \sqrt{3 \times 5 \times 17}
\]
This is already the simplest radical form. The decimal value is:
\[
\sqrt{255} \approx 15.969
\]

**Final answer:** \( \sqrt{255} \approx 15.969 \)

### Method 2: Long Division
Group the digits of 255 in pairs from the right: 2 and 55.
Find the largest number whose square is at most 22: that is 1, since 1^2 = 1.

Subtract to get remainder 1, then bring down 55 to make 155.
Double the quotient so far (1) to get 2; find a digit x so that 2x \times x ≤ 155. Test x=5: 25×5=125≤155. The quotient is now 15, remainder 155−125=30; place a decimal point and bring down 0000 to make 3000.

Continue the process to reach 15.96…

**Final answer:** \( \sqrt{255} \approx 15.969 \)

## Common Mistakes With The Square Root of 255

### Mistake 1: Trying to simplify a number with no square factor
**Correct way:** Factor first: 255 has no repeated prime, so \( \sqrt{255} \) is already simplest.

### Mistake 2: Rounding 255 up to 256 and reading 16
**Correct way:**  \( 16^2 = 256 \neq 255 \), so \( \sqrt{255} < 16 \).

### Mistake 3: Doubling The Wrong Number in Long Division
**Correct way:** Double the current quotient each step to build correctly for the next divisor.

## A Quick Way To Check Yourself
Estimate first: since 255 sits between 225 and 256, the root must be between 15 and 16, and because 255 is so close to 256, the answer should be near 16.

## Frequently Asked Questions
1. **Can the square root of 255 be simplified?**
   No, \( \sqrt{255} \) is already in simplest form.

2. **Is the square root of 255 rational or irrational?**
   Irrational. 255 is not a perfect square.

3. **What is the square root of 255 to three decimal places?**
   \( \sqrt{255} \approx 15.969 \).

4. **Is 255 a perfect square?**
   No, it is just below 256.

5. **What is the square root of 256?**
   Exactly 16.