Square Root of 1000 — Value, Simplification, and Steps

The square root of 1000 is approximately 31.623, and its exact simplified form is $10\sqrt{10}$.

Quick Answer:

Result: $\sqrt{1000} = 10\sqrt{10} \approx 31.623$

Notation: Simplified radical $10\sqrt{10}$; decimal $\approx 31.623$ (irrational, non-terminating)

Method shown: Prime factorization to simplify the radical, plus estimation between perfect squares

Approximate value: $31.6227766...$

Exact form: $10\sqrt{10}$ (never a whole number, since 1000 is not a perfect square)

Quick Reference Table

Notice the last row: because $\sqrt{1000} = 10\sqrt{10}$, its decimal is exactly ten times $\sqrt{10} \approx 3.162$.

Where the Square Root of 1000 Appears

The number 1000 is $10^3$, so $\sqrt{1000}$ is the same as $10^{3/2}$, which shows up whenever a quantity scales with the three-halves power of a length. It also appears in the diagonal of a box measuring $10 \times 10 \times 10\sqrt{8}$ and in decibel work, where a power ratio of 1000 corresponds to 30 dB. Any time you compute the geometric mean of two numbers whose product is 1000, the answer is $\sqrt{1000}$.

What a Square Root Means Here

A square root of a number is the value that, multiplied by itself, gives that number. Since no whole number times itself equals 1000 ($31^2 = 961$ and $32^2 = 1024$), the square root of 1000 is irrational, meaning its decimal never terminates or repeats. That is why we keep an exact form, $10\sqrt{10}$, alongside the rounded decimal.

How to Find the Square Root of 1000

Method 1: Prime factorization (gives the exact simplified form)

Break 1000 into primes.

$1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5$

Pair the identical factors.

$1000 = (2 \times 2) \times (5 \times 5) \times (2 \times 5)$

Each complete pair comes out of the radical as a single factor.

$\sqrt{1000} = 2 \times 5 \times \sqrt{2 \times 5}$

$\sqrt{1000} = 10\sqrt{10}$

Final answer: $10\sqrt{10}$

Method 2: Largest perfect-square factor (a faster shortcut)

Find the biggest perfect square that divides 1000. That is 100.

$1000 = 100 \times 10$

Split the radical across the product.

$\sqrt{1000} = \sqrt{100} \times \sqrt{10}$

$\sqrt{1000} = 10\sqrt{10}$

Final answer: $10\sqrt{10}$

Method 3: Estimating the decimal by hand

Locate 1000 between two perfect squares.

$31^2 = 961 \quad \text{and} \quad 32^2 = 1024$

So the root sits between 31 and 32, closer to 32 because 1000 is nearer 1024.

Test 31.6.

$31.6^2 = 998.56$

Test 31.63.

$31.63^2 = 1000.46$

The root is between 31.62 and 31.63, giving $\sqrt{1000} \approx 31.623$.

Final answer: $\approx 31.623$

Common Mistakes With Square Root of 1000

Mistake 1: Simplifying to $10\sqrt{100}$ instead of $10\sqrt{10}$

Where it slips in: Splitting 1000 as $100 \times 10$ but then pulling the wrong factor.

Don't do this: Writing $\sqrt{1000} = 10\sqrt{100}$, which equals $10 \times 10 = 100$ (far too big).

The correct way: The perfect square 100 comes out as 10, and the remaining 10 stays under the radical: $10\sqrt{10}$.

Mistake 2: Rounding too early and calling it exact

Where it slips in: Reporting $31.62$ as if it were the final answer.

Don't do this: Treating a rounded decimal as equal to $\sqrt{1000}$; the value is irrational and never stops.

The correct way: Keep $10\sqrt{10}$ for exact work and use $\approx 31.623$ only when a decimal is needed.

Mistake 3: Leaving a factor behind under the radical

Where it slips in: Stopping at $\sqrt{1000} = 5\sqrt{40}$ and thinking you are done.

The correct way: Check whether the number under the radical still has a perfect-square factor. Here $40 = 4 \times 10$, so keep simplifying until only 10 remains: $10\sqrt{10}$.

Conclusion

• The square root of 1000 is $10\sqrt{10}$ in exact form and about $31.623$ as a decimal.

• Prime factorization ($2^3 \times 5^3$) or the largest-perfect-square shortcut ($100 \times 10$) both give $10\sqrt{10}$.

• Because 1000 is not a perfect square, the root is irrational — keep the radical for exact answers.

To build fluency with radicals like this, work through problems with an algebra tutor or join structured math classes online.

Read More

• How to simplify the square root of 360

• Square root of 10 and its geometric meaning

• What is a square root — definition and method

• Squares and square roots explained

• Perfect squares and how to spot them