Square Root of 300

√300 at a glance

Exact value

10√3

Decimal (10 places)

17.3205080757

Rounded

17.3 · 17.32 · 17.321

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.320508

Prime factorization

2² × 3 × 5²

Cube root

6.694330

How to simplify √300

Look for the largest perfect square that divides 300. Here it is 100 (10²), because 300 = 100 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 300 = 2² × 3 × 5². Each pair of equal primes leaves the radical as one factor, so 2 × 5 comes out and 3 stays inside.

300 has 3 square factors (4, 25 and 100). Starting with a smaller one still works but takes more rounds: √300 = 2√75, and √75 can be simplified again. Using 100 straight away finishes in one step.

Check: (10√3)² = 10² × 3 = 100 × 3 = 300. As a decimal, 10√3 = 10 × 1.7320508076 ≈ 17.3205080757.

Where √300 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √300 lies between 17 and 18. 300 is 11 above 289 and 24 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.3143 (0.04% low)
• Tangent from 17, i.e. 17 + 11 ÷ 34: 17.3235 (0.02% high)
• Tangent from 18, i.e. 18 − 24 ÷ 36: 17.3333 (0.07% high)

For √300 the tangent at 17 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 300 is just 11 above 289.

Finding √300 with the Babylonian method

Picture a rectangle with an area of 300 and one side x; the other side must be 300 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √300.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √300 = 17.3205080757 to every decimal shown.

√300 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √300 the pattern is [17; 3, 8, 3, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √300 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 300y² = 1. Its smallest solution in positive whole numbers is x = 1,351, y = 78.

√300 in geometry and everyday measurements

• A square patio or deck of 300 square feet is about 17.32 ft (17 ft 4 in) on each side, so edging all the way around takes 4 × √300 ≈ 69.3 ft.
• 300 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √300 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 10 × 14 box, because 2² + 10² + 14² = 300.
• Since √300 = 10√3, a length of √300 is exactly 10 copies of the length √3 laid end to end.

Square roots near √300 and related roots

• The cube root of 300 is about 6.694330.
• Dividing by 100 divides the root by 10: √3 = √300 ÷ 10 ≈ 1.73205081.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.