Square Root of 255 — Value, Simplification, and Steps

The square root of 255 is approximately 15.969, and it stays as $\sqrt{255}$ because $255$ has no perfect-square factor to pull out.

Quick Answer:

Result: $\sqrt{255} \approx 15.969$

Notation: $\sqrt{255}$ (already in simplest radical form)

Method shown: Prime factorization and long division

Approximate value (irrational): $15.968719423$

Exact form: $\sqrt{255}$ (cannot be simplified further)

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 $256 = 16^2$, 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 $16 \times 16$ 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:

$\sqrt{255} = \sqrt{3 \times 5 \times 17}$

$\sqrt{255} = \sqrt{255}$

This is already the simplest radical form. The decimal value is:

$\sqrt{255} \approx 15.969$

Final answer: $\sqrt{255} \approx 15.969$ (exact form $\sqrt{255}$)

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 $2$: 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 $2x \times x \le 155$.

Test $x = 5$: $25 \times 5 = 125 \le 155$; test $x = 6$: $26 \times 6 = 156 > 155$. So $x = 5$.

The quotient is now $15$, remainder $155 - 125 = 30$; place a decimal point and bring down $00$ to make $3000$.

Continue the process to reach $15.96\ldots$

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

Where it slips in: Assuming every large number under a radical must simplify.

Don't do this: Force $\sqrt{255}$ into some $a\sqrt{b}$ form by guessing.

The correct way: Factor first: $255 = 3 \times 5 \times 17$ has no repeated prime, so $\sqrt{255}$ is already simplest. The first instinct is to hunt for a factor to pull out, but not every root simplifies — some numbers just stay under the radical.

Mistake 2: Rounding 255 up to 256 and reading 16

Where it slips in: Noticing $255$ is one less than $256 = 16^2$.

Don't do this: Write $\sqrt{255} = 16$.

The correct way: $16^2 = 256 \ne 255$, so $\sqrt{255} < 16$; the value is $15.969$, close to but not equal to $16$.

Mistake 3: Doubling The Wrong Number in Long Division

Where it slips in: After finding the first digit, deciding the next divisor.

Don't do this: Add the digit to itself as $1 + 1 = 2$ only sometimes, or double the whole running quotient inconsistently.

The correct way: Double the current quotient each step to build the new divisor's leading digits — for a quotient of $1$ the divisor starts at $2$, for $15$ it starts at $30$.

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 $16$. To build this fluency with a teacher, explore Bhanzu's algebra tutor or structured help with algebra.

Read More

• Square root — what a square root is and how the symbol works.

• Square root 1 to 25 — the reference chart for the small roots.

• Square root of 153 — simplifies to $3\sqrt{17}$.

• Square root of 43 — a prime, irrational root.

• Square root of 567 — simplifies to $9\sqrt{7}$.

• Squares and square roots — the wider topic.