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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

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.