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
Number $n$ | $\sqrt{n}$ exact | $\sqrt{n}$ approx |
|---|---|---|
900 | $30$ | $30.000$ |
1000 | $10\sqrt{10}$ | $31.623$ |
1024 | $32$ | $32.000$ |
1100 | $10\sqrt{11}$ | $33.166$ |
1200 | $20\sqrt{3}$ | $34.641$ |
2000 | $20\sqrt{5}$ | $44.721$ |
360 | $6\sqrt{10}$ | $18.974$ |
10 | $\sqrt{10}$ | $3.162$ |
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.
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