Square Root of 512 - Value, Simplified Form, and How to Find It

What Is a Square Root?

The square root of a number $n$ is the value $r$ for which $r^2 = n$. So $\sqrt{512}$ is the number that, times itself, returns 512.

No whole number works: $22^2 = 484$ is too small and $23^2 = 529$ is too big. That fixes $\sqrt{512}$ between 22 and 23.

Where Does the Square Root of 512 Appear?

$\sqrt{512}$ shows up as a diagonal, because a square with side 16 has a diagonal of exactly $16\sqrt{2}$, which is $\sqrt{512}$. It is also close to home for anyone in computing, where $512 = 2^9$ is a familiar block size, 512 bytes is the classic disk sector, and its square root inherits that pure power-of-two structure, collapsing to the tidy $16\sqrt{2}$.

Quick Reference Table

Is 512 a Perfect Square?

No. A perfect square is an integer times itself, 1, 4, 9, 16, 25, and so on. 512 is a perfect cube ($8^3 = 512$), but it is not a perfect square.

Because 512 is not a perfect square, its square root is not a whole number or a fraction. That is what makes $\sqrt{512}$ irrational.

Is the Square Root of 512 Rational or Irrational?

$\sqrt{512}$ is irrational, it cannot be written as a fraction $\frac{p}{q}$ of integers, and its decimal neither ends nor repeats. A whole number has a rational square root only when it is a perfect square, and 512 is not.

Reducing it to $16\sqrt{2}$ changes nothing here. The surviving $\sqrt{2}$ is irrational, so the product is too, a fact you can trace back to the formal definition of a square root.

What Is the Square Root of 512 in Simplest Radical Form?

The simplest radical form of $\sqrt{512}$ is $16\sqrt{2}$. Because $512 = 2^9$ is a pure power of two, simplifying the radical expression is especially clean.

Prime factorization method.

$512 = 2^9$

$\sqrt{512} = \sqrt{2^8 \times 2}$

$\sqrt{512} = 2^4 \times \sqrt{2}$

$\sqrt{512} = 16\sqrt{2}$

The eight paired 2s step out as $2^4 = 16$; a single 2 is left over, so it stays inside as $\sqrt{2}$. That leftover is the same $\sqrt{2}$ you find in the square root of 8, which reduces to $2\sqrt{2}$.

How Do You Find the Square Root of 512 by Long Division?

Long division produces the decimal digits in order. Here is the value to two decimals, one step per line.

Pair the digits from the decimal point: $\overline{5},\overline{12}.\overline{00},\overline{00}$

Largest square $\leq 5$ is $2^2 = 4$, so the first digit is 2, remainder 1.

Bring down 12: the value is 112; double the quotient (2) to get 4, and $42 \times 2 = 84 \leq 112$, so the next digit is 2, remainder 28.

Bring down 00: the value is 2800; double 22 to get 44, and $446 \times 6 = 2676 \leq 2800$, so the next digit is 6, remainder 124.

Bring down 00: the value is 12400; double 226 to get 452, and $4522 \times 2 = 9044 \leq 12400$, so the next digit is 2.

That gives $\sqrt{512} \approx 22.62$, and continuing yields $22.6274$. The digits never repeat, which is the mark of an irrational number.

Examples Of the Square Root of 512

Example 1

Simplify $\sqrt{512}$ to its radical form.

$512 = 2^9$

$\sqrt{512} = 2^4 \times \sqrt{2} = 16\sqrt{2}$

Final answer: $16\sqrt{2}$

Example 2

A student simplifies $\sqrt{512}$ and stops at $\sqrt{512} = 2\sqrt{128}$. Is that the simplest form?

The first instinct is to pull out the smallest easy factor and move on. Taking out 4 gives $\sqrt{512} = 2\sqrt{128}$, which is true but not simplest.

Check the leftover: $128 = 64 \times 2$ still hides a perfect square. So $2\sqrt{128} = 2 \times 8\sqrt{2} = 16\sqrt{2}$.

Final answer: $16\sqrt{2}$, keep going until the radicand has no square factor left.

Example 3

Evaluate $(\sqrt{512})^2$.

$(\sqrt{512})^2 = 512$

Final answer: $512$

Example 4

A square has area 512 square units. Find its side length.

side $= \sqrt{512}$

side $= 16\sqrt{2} \approx 22.6274$

Final answer: $16\sqrt{2}$ units, about $22.63$ units.

Example 5

Simplify $\sqrt{512} - \sqrt{128}$.

$\sqrt{512} - \sqrt{128} = 16\sqrt{2} - 8\sqrt{2}$

$\sqrt{512} - \sqrt{128} = 8\sqrt{2} \approx 11.314$

Final answer: $8\sqrt{2}$. Both terms reduce to a multiple of $\sqrt{2}$, so they subtract like ordinary like terms.

Common Mistakes

Mistake 1: Stopping at the first perfect-square factor

Where it slips in: Simplifying $\sqrt{512}$ by pulling out only 4.

Don't do this: Writing $\sqrt{512} = 2\sqrt{128}$ and treating it as done.

The correct way: Factor until nothing square remains: $2\sqrt{128} = 16\sqrt{2}$. When the number is a high power of two, the first-instinct error is almost always stopping too early.

Mistake 2: Confusing the square root with the cube root

Where it slips in: Remembering that $8^3 = 512$ and reporting 8.

Don't do this: Writing $\sqrt{512} = 8$ because 512 is a perfect cube.

The correct way: $8$ is the cube root, not the square root. The square root is $16\sqrt{2} \approx 22.63$; the two operations answer different questions.

Mistake 3: Rounding too early

Where it slips in: Multi-step problems that reuse $\sqrt{512}$.

Don't do this: Replacing $\sqrt{512}$ with $22.6274$ at the very start.

The correct way: Keep $16\sqrt{2}$ exact until the final step, then round once. Carrying a rounded value from the start lets error compound through every later operation.

Conclusion

The square root of 512 is $16\sqrt{2}$, roughly $22.6274$, and it is irrational because 512 has no whole-number square root. Write $512 = 2^9$, step every pair of 2s out, and keep the exact radical until the final line. To build these radical and exponent skills with a teacher, explore Bhanzu's algebra tutor, join live math classes online, or work with a high school math tutor.

Want to practice with a guide? Book a free demo class and simplify radicals step by step.

Read More

• Square Root 1 to 30, the full reference table of roots from 1 to 30.

• Square root tricks, quick ways to estimate and simplify roots.

• Square Root of 1000, another large radical worth simplifying.

• Square Root of 2000, a related root with the same $\sqrt{2}$ leftover pattern.

• Squares and square roots, how perfect squares connect to their roots.