Square Root of 1000 Simplified

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What is the square root of 1,000 simplified?

Answer

√1,000 = 10√10

1,000 = 100 × 10, and 100 = 10² is the largest perfect square that divides 1,000, so √1,000 = √100 × √10 = 10√10. Check: 10√10 ≈ 31.623, the same as √1,000 ≈ 31.623.

Simplifying √1,000 Step by Step

1. Prime factorization. — All Prime Factors of 1,000:

1,000 = 2 × 2 × 2 × 5 × 5 × 5 = 23 × 53

2. Pair up equal primes. Each pair is a perfect square and comes out of the root as one factor; a prime without a partner stays under the radical.

Prime factorTimes in 1,000Comes out of the rootStays under the root
2322
5355

3. Take the pairs out of the root.

√1,000 = √((2 × 2) × (5 × 5) × 2 × 5)

= 2 × 5 × √(2 × 5)

= 10√10

4. Check by squaring back.

10² × 10 = 100 × 10 = 1,000

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Simplify Square Root Calculator

This calculator writes a square root in simplest radical form: it finds the largest perfect square that divides the number, takes its root out of the radical and leaves the rest under it. For example, it can help you find out what is the square root of 1,000 simplified? (The answer is: 10√10). Enter a whole number (e.g. '1,000') and hit the 'Simplify' button.
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Simplified Square Roots Table

NumberSimplest radical formDecimal value
67515√3≈ 25.981
72012√5≈ 26.833
7255√29≈ 26.926
72611√6≈ 26.944
7292727
7505√30≈ 27.386
7566√21≈ 27.495
7722√193≈ 27.785
7802√195≈ 27.928
80020√2≈ 28.284
8505√34≈ 29.155
8755√35≈ 29.58
8804√55≈ 29.665
9608√15≈ 30.984
97218√3≈ 31.177
1,00010√10≈ 31.623
1,20020√3≈ 34.641
1,25025√2≈ 35.355
1,2606√35≈ 35.496
1,28016√5≈ 35.777
1,2963636
1,57515√7≈ 39.686
1,87525√3≈ 43.301
2,00020√5≈ 44.721
2,01612√14≈ 44.9
3,7806√105≈ 61.482
5,00050√2≈ 70.711
7,20060√2≈ 84.853
8,00040√5≈ 89.443
12,50050√5≈ 111.803

FAQ

What is the square root of 1,000 simplified?

√1,000 = 10√10

Is 1,000 a perfect square?

No. 1,000 = 100 × 10: its largest perfect square factor is 100 = 10², and the factor 10 stays under the radical, so √1,000 = 10√10.

What is √1,000 as a decimal?

√1,000 ≈ 31.623. The decimal never ends and never repeats, so it is always rounded; the exact value is 10√10.

What is the largest perfect square that divides 1,000?

100 = 10². Dividing gives 1,000 ÷ 100 = 10, and 10 has no square factor left, so √1,000 = √100 × √10 = 10√10.

See Also